# Skew analysis over quaternions. I

Masood Aryapoor
Division of Mathematics and Physics
Mälardalen University
Hamngatan 15, 632 17, Eskilstuna, Sweden

###### Abstract

We introduce a class of rings using which we define the concept of skew
regularity for quaternion-valued functions over quaternions. It is shown that
the notion of skew regularity coincides with the concept of slice regularity
over symmetric slice domains. Known results regarding slice-regular functions
over symmetric slice domains are generalized to skew-regular functions over
general symmetric domains. Furthermore, we present new results concerning
slice-regular functions.

##  1 Introduction

This paper consists of two parts: An algebraic part and an analytical part. In
the algebraic part, we introduce and study a class of rings which we shall
call skew-convex function rings. They can be regarded as generalizations of
skew polynomial rings over skew fields. In the analytical part, we use skew-
convex function rings over the skew field of quaternions to develop an
analytical theory over quaternions which will be called skew analysis over
quaternions. It turns that skew analysis over quaternions is closely related
to the theory of slice-regular functions as introduced in [6] and later
developed in a series of papers. For a detailed account of slice-regular
functions, see [5] and references therein.

In Section 2, we introduce the notion of the skew product which is a binary
operation on certain sets of functions with values in a fixed skew field (see
Definition 2.1). The concept of the skew product is motivated by the product
formula for the evaluation of skew polynomials introduced in [10, Theorem
2.7]. We show that the skew product gives rise to near-ring structures on sets
of functions with values in the skew field. Restricting our attention to the
class of “skew-convex” functions, we arrive at the notion of skew-convex
function rings (see Definition 2.2 and Theorem 2.4). In the rest of Section 2,
we study invertible elements in skew-convex function rings. A more complete
treatment of the algebraic properties of skew-convex function rings will
appear elsewhere.

Section 3 deals with skew analysis over quaternions. In this section, the
notion of skew regularity is introduced (see Definition 3.2) using which we
develop an analytical theory for quaternion-valued functions over quaternions.
In particular, we define the notion of the skew derivative which is very
similar to the classical definition of derivatives (see Definition 3.3). It
turns out that skew-regular functions are always skew-convex, a fact with far-
reaching consequences. As mentioned above, the concept of skew regularity is
closely related to the notion of slice regularity. More precisely, we show
that every skew-regular function is slice regular (see Proposition 3.19). More
importantly, we will prove that a function defined on a “symmetric slice
domain” is skew regular if and only if it is slice regular (see Subsection 3.5
and Theorem 3.20). It is known that the notion of slice regularity is not
well-behaved over domains that are not slice domains. The strength of our
approach lies in the fact that the concept of skew-regularity is well-behaved
even over domains which are not slice domains. One of our main results is that
over a general symmetric domain, a function is skew regular if and only if it
is symmetrically analytic (see Theorem 3.26). Furthermore, our approach
simplifies and clarifies some of the results and features of slice-regular
functions.

##  2 Rings of skew maps

The theory of skew polynomial rings over skew fields is rich and well-
developed. For a detailed account of this theory, see for example [1, Section
1.1]. In [10], the authors defined the evaluation of skew polynomials using
which skew polynomials can naturally be considered as maps on the ground skew
field. It turns out that the value of the product of two skew polynomials at a
given point is not necessarily equal to the product of the values of the skew
polynomials at the same point. The correct formula, called the product
formula, is given in [10, Theorem 2.7]. The product formula can be regarded as
a binary operation which we shall call the skew product. This section will
deal with the skew product and its properties. Subsection 2.1 introduces the
notions of the skew product and skew-convex functions, and gives their basic
properties. In Section 2.2, we study skew-invertible functions, that is,
functions which are invertible with respect to the skew product.

###  2.1 Rings of skew-convex functions

Let $K$ be a skew field and $X$ be a set on which the multiplicative group
$K^{*}\colonequals K\setminus\\{0\\}$ acts (on the left). The action of $a\in
K^{*}$ on $x\in X$ is denoted by ${}^{a}x$. We will freely use the standard
concepts from Group Theory. In particular, we use the following notions: An
invariant subset of $X$ is a subset $Y$ of $X$ which satisfies the property
that $a\in K^{*}$ and $y\in Y$ imply that ${}^{a}y\in Y$; By an orbit, we mean
the smallest invariant subset containing some element.

We denote the set of all functions $f\colon X\to K$ by $\mathcal{F}(X)$. By
abuse of notation, a constant function in $\mathcal{F}(X)$, whose value is
$a\in K$, is simply denoted by $a$. Given functions $f,g\colon X\to K$, let
$f+g$ denote the pointwise addition of the functions $f$ and $g$.

######  Definition 2.1.

The left skew product of the functions $f$ and $g$ as follows

$(f\diamond g)(a)=\begin{cases}f\left(\,{}^{{g(a)}}a\,\right)g(a)&g(a)\neq
0,\\\ 0&g(a)=0.\end{cases}$ (2.1)

The right skew product is defined as follows

$(f\diamond_{r}g)(a)=\begin{cases}f(a)g\left(\,{}^{{f(a)^{-1}}}a\,\right)&f(a)\neq
0,\\\ 0&f(a)=0.\end{cases}$ (2.2)

In this paper, we will mostly work with the left skew product. Therefore, we
usually drop the adjective “left” if there is no risk of confusion. We leave
it to the reader to formulate and prove similar results for the right skew
product.

It is easy to see that $(a\diamond f)(x)=af(x)$, for every $a\in K,x\in X$. We
shall henceforth denote $a\diamond f$ by $af$. In the following lemma, the
proof of which is straightforward, we collect some properties of the skew
product.

######  Lemma 2.1.

Let $f,h,g\colon X\to K$ be arbitrary functions. Then:
(1) The constant function $1\colon x\mapsto 1$ is a unit for $\diamond$, that
is, $f=f\diamond 1=1\diamond f$.
(2) $(f+g)\diamond h=f\diamond h+g\diamond h$, that is, the right distributive
law (with respect to pointwise addition) holds for the skew product.
(3) $(f\diamond g)\diamond h=f\diamond(g\diamond h)$, that is, $\diamond$ is
associative.

It follows from this lemma that the set $\mathcal{F}(X)$ equipped with
pointwise addition and the skew product is a structure known as right near-
ring in the literature. We note that the skew product may not be left
distributive with respect to pointwise addition. However, the left
distributive law for $\diamond$ holds for a class of functions described
below.

######  Definition 2.2.

A function $f\colon X\to K$ is called skew convex if

$f\diamond(a+b)=f\diamond a+f\diamond b,\text{ for all }a,b\in K.$

The set of all skew-convex functions $f\colon X\to K$ is denoted by
$\mathcal{S}(X)$.

Any constant function belongs to $\mathcal{S}(X)$ since $a\diamond b=ab$ for
all $a,b\in K$. More generally, we have the following result. The proof is
left to the reader.

######  Proposition 2.2.

A function $f\colon X\to K$ which is constant on every orbit in $X$ is skew
convex.

The following lemma justifies the introduction of $\mathcal{S}(X)$.

######  Lemma 2.3.

Let $h\colon X\to K$ be given. The condition

$h\diamond(f+g)=h\diamond f+h\diamond g,$

holds for all functions $f,g\colon X\to K$ if and only if $h$ is skew convex.

###### Proof.

The result follows from the identity

$(h\diamond f)(x)=(h\diamond{f(x)})(x),\text{ for all
}h,f\in\mathcal{F}(X)\text{ and }x\in X.$

∎

As a consequence of this lemma, we have the following result.

######  Theorem 2.4.

Equipped with the left skew product, the additive group $\mathcal{S}(X)$ is a
ring with identity.

###### Proof.

The result follows from Lemma 2.3 and the general fact that in any right near-
ring $R$, the set

$\\{r\in R\,|\,r(s_{1}+s_{2})=rs_{1}+rs_{2}\text{ for all }s_{1},s_{2}\in
R\\},$

is a ring. ∎

We call $\mathcal{S}(X)$, equipped with pointwise addition and the skew
product, the ring of skew-convex functions on $X$ determined by the action of
$K^{*}$ on $X$. To introduce a subring of $\mathcal{S}(X)$, let us denote the
set of all functions $f\colon X\to K$ which are constants on every orbit, by
$\mathcal{C}(X)$. By Proposition 2.2, we have
$\mathcal{C}(X)\subset\mathcal{S}(X)$. Moreover, we have the following
proposition whose proof is left to the reader.

######  Proposition 2.5.

The set $\mathcal{C}(X)$ is a subring of $\mathcal{S}(X)$. Moreover, it is
isomorphic to the ordinary ring of functions $f\colon X/K^{*}\to K$ (equipped
with pointwise addition and pointwise multiplication). Here, $X/K^{*}$ denotes
the set of equivalence classes of the action.

Let us now give some examples.

######  Example 2.1.

If the action of $K^{*}$ on $X$ is trivial, the ring $\mathcal{S}(X)$ is just
the ring of all functions $f\colon X\to K$ equipped with pointwise addition
and pointwise multiplication.

######  Example 2.2.

Let $\sigma\colon K\to K$ be an endomorphism and $\delta\colon K\to K$ be a
$\sigma$-derivation. Let $K[T;\sigma,\delta]$ denote the ring of skew
polynomials determined by $\sigma$ and $\delta$. Every elements of
$K[T;\sigma,\delta]$ can uniquely be written as $\sum_{m=0}^{n}a_{m}T^{m}$
where $a_{m}\in K$ with $a_{n}\neq 0$. Recall that the identity
$Ta=\sigma(a)T+\delta(a)$, where $a\in K$, holds in $K[T;\sigma,\delta]$. One
can define the $(\sigma,\delta)$-action of $K^{*}$ on $K$ as follows

$^{b}a=\sigma(b)ab^{-1}+\delta(b)b^{-1}.$ (2.3)

The ring of skew-convex functions determined by this action is denoted by
$K[\sigma,\delta]$. One can verify that there exists a unique ring
homomorphism

$K[T;\sigma,\delta]\to K[\sigma,\delta],$

which sends each $a\in K$ to the constant function $a$, and $T$ to the
identity function $1_{K}\colon K\to K$. In particular, any skew polynomial
$P(T)\in K[T;\sigma,\delta]$ can, under this homomorphism, be considered as a
skew-convex function $P\colon K\to K$. The reader can verify that the value
$P(a)$ of $P$ at $a\in K$ coincides with the evaluation map introduced in
[10], that is, $P(a)$ is the unique element of $K$ for which we have

$P(T)-P(a)\in K[T;\sigma,\delta](T-a).$

In the particular case where $\delta$ is zero and $\sigma$ is the identity
homomorphism, the value $P(a)$ is computed as follows

$P(a)=\sum_{m=0}^{n}a_{m}a^{m},\text{ where }P(T)=\sum_{m=0}^{n}a_{m}T^{m}\in
K[T]\colonequals K[T;1_{K},0].$

We conclude this part with a discussion regarding pullbacks. Let $Y$ be a
second set on which $K^{*}$ acts (on the left). For a map $\phi\colon X\to Y$,
let $\phi^{*}\colon\mathcal{F}(Y)\to\mathcal{F}(X)$ denote the pullback map
defined by $\phi^{*}(f)=f\circ\phi$. Recall that a map $\phi\colon X\to Y$ is
called action-preserving if $\phi(^{a}x)={}^{a}\phi(x)$ for all $a\in K^{*}$
and $x\in X$.

######  Proposition 2.6.

Suppose that $\phi\colon X\to Y$ is action-preserving. Then, for any function
$f\in\mathcal{S}(Y)$, we have $\phi^{*}(f)\in\mathcal{S}(X)$. Moreover, the
map $\phi^{*}\colon\mathcal{S}(Y)\to\mathcal{S}(X)$ is a homomorphism of
rings.

###### Proof.

The fact that $\phi^{*}(f)\in\mathcal{S}(X)$ for $f\in\mathcal{S}(Y)$, follows
from the identity

$(f\circ\phi)\diamond a=(f\diamond a)\circ\phi,\text{ where }a\in K.$

The rest of the proof is straightforward. ∎

In particular, if $Z\subset X$ is an invariant subset of $X$, then the
restriction map $f\in\mathcal{S}(X)\mapsto f|_{Z}\in\mathcal{S}(Z)$ is a well-
defined homomorphism of rings.

###  2.2 Skew-invertible functions

As before, let $K$ be a skew field and $X$ be a set on which $K^{*}$ acts. A
function $f\in\mathcal{F}(X)$ is called skew invertible if it is invertible
with respect to the skew product, in which case, the inverse of $f$ is called
its skew inverse and denoted by $f^{\langle-1\rangle}$. The notion of skew
inverse with respect to the right skew product is defined in the obvious way.
The skew inverse of a function $f$ with respect to the right skew product will
be denoted by $f^{\langle-1\rangle_{r}}$.

######  Lemma 2.7.

If $f\in\mathcal{S}(X)$ is skew invertible, then the skew inverse of $f$
belongs to $\mathcal{S}(X)$.

###### Proof.

By Lemma 2.3, we have

$\mathcal{S}(X)=\\{f\in\mathcal{F}(X)\,|\,f\diamond(g+h)=f\diamond g+f\diamond
h\text{ for all }g,h\in\mathcal{F}(X)\\}.$

The result follows from the following general fact whose proof is left to the
reader: Let $R$ be a right near-ring and consider the ring

$R^{\prime}=\\{r\in R\,|\,r(s_{1}+s_{2})=rs_{1}+rs_{2}\text{ for all
}s_{1},s_{2}\in R\\}.$

If $r\in R^{\prime}$ is invertible in $R$, then its inverse belongs to
$R^{\prime}$. ∎

Next, we give a characterization of skew-invertible functions.

######  Lemma 2.8.

(1) Let $f\in\mathcal{F}(X)$. There exists $g\in\mathcal{F}(X)$ such that
$f\diamond g=1$ if and only if for every $x\in X$, there exists some $a\in
K^{*}$ such that $f({}^{a}x)=a^{-1}$.
(2) Let $g\in\mathcal{F}(X)$. There exists $f\in\mathcal{F}(X)$ such that
$f\diamond g=1$ if and only if $g(X)\subset K^{*}$, and the map
$x\mapsto{}^{g(x)}x$ is 1-1.
(3) A function $f\in\mathcal{F}(X)$ is skew invertible if and only if
$f(X)\subset K^{*}$ and the assignment $x\mapsto{}^{f(x)}x$ establishes a
bijection from $X$ onto $X$.

###### Proof.

(1) The proof is straightforward.
(2) First, we prove the “if” direction. It follows from $f\diamond g=1$ that
$f({}^{g(x)}x)g(x)=1$ for all $x\in X$. Clearly, $g$ must be nonzero on $X$,
i.e., $g(X)\subset K^{*}$. If ${}^{g(x)}x={}^{g(y)}y=z\in K$ for some $x,y\in
X$, then we have $f(z)g(x)=f(z)g(y)=1$, implying that $g(x)=g(y)$. One can
easily verify that ${}^{g(x)}x={}^{g(y)}y$ and $g(x)=g(y)$ imply that $x=y$.
This proves the “if” direction. Conversely, let $g$ satisfy the stated
properties. We define a map $f\colon X\to K$ as follows: If $y={}^{g(x)}x$ for
some $x\in X$, we set $f(y)=g(x)^{-1}$. Otherwise, we set $f(y)=x_{0}$ where
$x_{0}\in X$ is a fixed element. It is easy to see that $f$ is well-defined
and $f\diamond g=1$.
(3) Suppose that $f$ is skew invertible and let $g$ be its skew inverse. By
$(2)$, $f$ is nonzero on $X$ and $x\mapsto{}^{f(x)}x$ is 1-1. The fact that
$x\mapsto{}^{f(x)}x$ is onto follows from the following identity

${}^{f(^{g(y)}y)}\left({}^{g(y)}y\right)=y.$

Conversely, suppose that $f$ satisfies the stated properties. By Part $(2)$,
there exists $g\in\mathcal{F}(X)$ such that $g\diamond f=1$. We need only show
that $f\diamond g=1$. Given an arbitrary element $x\in X$, we can choose $y\in
X$ such that $x={}^{f(y)}y$. We have

$(g\diamond f)(y)=1\implies g({}^{f(y)}y)f(y)=1\implies g(x)f(y)=1.$

Therefore, we have ${}^{g(x)}x={}^{g(x)f(y)}y=y$ from which it follows that

$g(x)f(y)=1\implies f(y)g(x)=1\implies f({}^{g(x)}x)g(x)=1\implies(f\diamond
g)(x)=1.$

Since $x\in X$ was arbitrary, we conclude that $f\diamond g=1$. ∎

A skew-invertible function shares some properties with its skew inverse. In
the following proposition, we give two such properties.

######  Proposition 2.9.

Let $f\colon X\to K^{*}$ be a skew-invertible function. Then:
(1) If $f$ is 1-1, so is $f^{\langle-1\rangle}$.
(2) If $f$ is onto, so is $f^{\langle-1\rangle}$.

###### Proof.

By Part (3) of Lemma 2.8, the function $\alpha\colon X\to X$, defined by
$\alpha(x)={}^{f(x)}x$, is a bijection. Let $\beta\colon X\to X$ denote its
inverse (with respect to function composition). It is easy to check that

$f^{\langle-1\rangle}(x)=f(\beta(x))^{-1},\text{ for all }x\in X,$

from which the results follow. ∎

Regarding skew-invertible elements of $\mathcal{S}(X)$, we have the following
characterization.

######  Proposition 2.10.

A skew-convex function $f:X\to K$ satisfying $f(X)\subset K^{*}$ is skew
invertible if and only if for any $x\in X$, there exists some $a\in K^{*}$
such that $f({}^{a}x)=a^{-1}$.

###### Proof.

The “only if” direction follows from Part 1 of Lemma 2.8. To prove the
converse, we use Part (2) of Lemma 2.8. Therefore, we need only show that if
${}^{f(x)}x={}^{f(y)}y$, then $x=y$. Let ${}^{f(x)}x={}^{f(y)}y$ for some
$x,y\in X$. Then $y$ and $x$ are in the same orbit, implying that there exists
$a\in K^{*}$ such that $y={}^{a}x$. So,

${}^{f(x)}x={}^{f(y)}y={}^{f({}^{a}x)a}x.$

It follows that $f(x)b=f({}^{a}x)a$ for some $b\in K^{*}$ satisfying
${}^{b}x=x$. Note that $f({}^{-a}x)=f({}^{a}x)$ because

$0=f\diamond(a+(-a))=f\diamond a+f\diamond(-a)\implies
f\diamond(-a)=-f\diamond a.$

Therefore, we can write

$0=f({}^{b}x)b-f({}^{-a}x)a=(f\diamond
b+f\diamond(-a))(x)=\left(f\diamond(b-a)\right)(x),$

implying that $a=b$. Since $f(X)\subset K^{*}$, we must have
$x={}^{b}x={}^{a}x=y$. ∎

We conclude the algebraic part of the paper with an example of a skew-
invertible function.

######  Example 2.3.

As in Example 2.2, let $\sigma\colon K\to K$ be an endomorphism and
$\delta\colon K\to K$ be a $\sigma$-derivation. Let $A\subset K$ be an
invariant set under the $(\sigma,\delta)$-action. Using Lemma 2.8 and
Proposition 2.10, we see that a linear polynomial $T-c\in K[T;\sigma,\delta]$,
where $c\in K$, is skew invertible as an element of $\mathcal{S}(A)$ if and
only if (1) $T-c$ does not have a root in $A$, and (2) for any $a\in A$, there
exists $b\in K^{*}$ such that $(T-c)({}^{b}a)=b^{-1}$. The first condition is
equivalent to $c\notin A$. The second condition is equivalent to the condition
that for any $a\in A$, there exists $b\in K^{*}$ satisfying the equation
${}^{b}a-c=b^{-1}$, or equivalently, the equation

$\sigma(b)a+\delta(b)-cb=1.$

Note that if $T-c$ is skew invertible, then for any $a\in A$, we have

$b=(T-c)^{\langle-1\rangle}(a)\iff\sigma(b)a+\delta(b)-cb=1.$

In the case where $\delta$ is zero and $\sigma$ is the identity homomorphism,
we have

$b=(T-c)^{\langle-1\rangle}(a)\iff ba-cb=1.$

##  3 Skew analysis over quaternions

A central problem of “quaternionic analysis” is to find a suitable notion of
“quaternionic derivative”. It is known that the most obvious definition,
namely,

$F^{\prime}(q_{0})=\lim_{q\to q_{0}}\left(F(q)-F(q_{0})\right)(q-q_{0})^{-1},$

is too restrictive. From the point of view of the skew product, one is led to
the informal definition

$F^{\prime}(q_{0})=\lim_{q\to
q_{0}}\left(\left(F-F(q_{0})\right)\diamond(T-q_{0})^{\langle-1\rangle}\right)(q),$

which we shall call the skew derivative (see Definition 3.3 for the precise
definition). This section deals with the notion of the skew derivative and
develops a version of quaternionic analysis based on the skew derivative.

Throughout this section, $\mathbb{H}$ denotes the skew field of quaternions.
Given a quaternion

$q=r_{0}+r_{1}i+r_{2}j+r_{3}k,\text{ where
}r_{0},r_{1},r_{2},r_{3}\in\mathbb{R},$

we denote the real part, imaginary part, conjugate and norm of $q$ by

$\text{Re}(q)=r_{0},\,\text{Im}(q)=r_{1}i+r_{2}j+r_{3}k,$
$\overline{q}=r_{0}-r_{1}i-r_{2}j-r_{3}k\text{ and
}|q|=\sqrt{r_{0}^{2}+r_{1}^{2}+r_{2}^{2}+r_{3}^{2}},\,\,\text{ respectively.}$

In this section, we fix the following action of $\mathbb{H}^{*}$ on
$\mathbb{H}$: ${}^{p}q=pqp^{-1}.$ The orbit containing a quaternion
$q\in\mathbb{H}$ is denoted by $O(q)$. It is known that for all
$p,q\in\mathbb{H}$, $O(p)=O(q)$ holds if and only if
$\text{Re}(p)=\text{Re}(q)$ and $|p|=|q|$ hold. In particular, an orbit is
either a single point, called a trivial orbit, or a 2-dimensional sphere,
called a nontrivial orbit. We also consider $\mathbb{H}$ as a topological
space where the topology is the Euclidean topology induced by the norm
$q\mapsto|q|$. For convenience, we shall use the following terminology: A
subset of $\mathbb{H}$ is called a region if it is both open and invariant
under the action ${}^{p}q=pqp^{-1}.$

###  3.1 Skew-convex functions on $\mathbb{H}$

In this section, we study skew-convex functions on invariant subsets of
$\mathbb{H}$. We begin with a characterization of skew-convex functions on
orbits.

######  Proposition 3.1.

Let $F\colon O\to\mathbb{H}$ be a function on an orbit $O$ of $\mathbb{H}$.
Then, $F$ is skew convex if and only if $F$ is (left) affine in the sense that
there exist constants $a,b\in\mathbb{H}$ such that

$F(q)=a+bq,\text{ for all }q\in O.$

###### Proof.

If $O$ is a trivial orbit, there is nothing to prove, so assume that $O$ is
nontrivial. The proof of the “if” direction is straightforward. To prove the
converse, fix $q_{0}\in O$. We need only show that

$(F(q_{1})-F(q_{0}))(q_{1}-q_{0})^{-1}=(F(q_{2})-F(q_{0}))(q_{2}-q_{0})^{-1},$

for all $q_{1}\neq q_{2}\in O\setminus\\{q_{0}\\}$. This identity can be
rewritten as

$F(q_{1})(q_{1}-q_{0})^{-1}-F(q_{2})(q_{2}-q_{0})^{-1}=F(q_{0})\left((q_{1}-q_{0})^{-1}-(q_{2}-q_{0})^{-1}\right).$
(3.1)

Since $q_{i}\in O(q_{0})$, we have

$q_{i}=(q_{i}-q_{0})^{-1}\overline{q}_{0}(q_{i}-q_{0}),\text{ for }i=1,2,$

using which the left-hand side of Identity 3.1 can be rewritten as

$F({}^{(q_{1}-q_{0})^{-1}}\overline{q}_{0})(q_{1}-q_{0})^{-1}-F({}^{(q_{2}-q_{0})^{-1}}\overline{q}_{0})(q_{2}-q_{0})^{-1}$
$=F({}^{\left((q_{1}-q_{0})^{-1}-(q_{2}-q_{0})^{-1}\right)}\overline{q}_{0})\left((q_{1}-q_{0})^{-1}-(q_{2}-q_{0})^{-1}\right).$

It is easy to check that
${}^{\left((q_{1}-q_{0})^{-1}-(q_{2}-q_{0})^{-1}\right)}\overline{q}_{0}=q_{0}$
form which Identity 3.1 follows. ∎

In view of this proposition, the following definition is well-defined.

######  Definition 3.1.

The orbital derivative of a skew-convex function $F:O\to\mathbb{H}$ at a
nontrivial orbit $O$ is defined to be the value

$\left(F(p)-F(q)\right)(p-q)^{-1},\text{ where }q\neq p\in O.$

The orbital derivative of $F$ at $O=O(q)$ is denoted by $F^{o}(O)$ or
$F^{o}(q)$. If $O$ is a trivial orbit, the orbital derivative of $F$ at $O$ is
defined to be zero.

Let us give another characterization of skew-convex functions which will be
needed later. In the following proposition, the function
$T\colon\mathbb{H}\to\mathbb{H}$ denotes the identity function on
$\mathbb{H}$.

######  Proposition 3.2.

Let $O\subset\mathbb{H}$ be a nontrivial orbit. A function $F\colon
O\to\mathbb{H}$ is skew convex on $O$ if and only if $F$ can be written as

$F=a+E\diamond(T-q),$

for some $a\in\mathbb{H}$, $q\in O$ and some function $E\colon
O\to\mathbb{H}$, in which case, $E(\overline{q})$ is equal to the orbital
derivative of $F$ at $O$.

###### Proof.

The “only if” direction follows directly from Proposition 3.1. To prove the
other direction, let $F=a+E\diamond(T-q)$, for some $a\in\mathbb{H}$, $q\in U$
and some function $E\colon O\to\mathbb{H}$. Evaluating at a quaternion
${}^{p}q$, where $p\in\mathbb{H}$ satisfies $pq\neq qp$, we have

$F({}^{p}q)=a+E({}^{({}^{p}q-q)p}q)({}^{p}q-q)=a+E({}^{pq-qp}q)({}^{p}q-q).$

Using the elementary formula

${}^{pq-qp}q=\overline{q},\text{ for all }p,q\in\mathbb{H}\text{ with }pq\neq
qp,$

we obtain $F({}^{p}q)=a+E(\overline{q})({}^{p}q-q)$, or equivalently,

$F(p)=a+E(\overline{q})(p-q),\text{ for all }p\in O.$

In particular, $F$ is skew convex. Note that $a=F(q)$ and

$E(\overline{q})=\left(F(p)-F(q)\right)(p-q)^{-1}=F^{o}(O),\text{ where }q\neq
p\in O.$

∎

In the following proposition, we collect some properties of the orbital
derivative.

######  Proposition 3.3.

Let $F,G\colon U\to\mathbb{H}$ be skew-convex functions on a region $U$ in
$\mathbb{H}$. Then, $F+G$ and $F\diamond G$ are skew convex on $U$, and
moreover, we have

$(F+G)^{o}=F^{o}+G^{o},\text{ on $U$ and },$ $(F\diamond
G)^{o}(p)=(F^{o}\diamond G)(p)+(F\diamond G^{o})(\overline{p}),\text{ for all
$p\in U$ }.$

###### Proof.

The first part follows from Theorem 2.4. The first formula is trivial. To
prove the second formula, let $p\in U$ be fixed and set $O=O(p)$. We assume
that $G(p)\neq 0$. The case $G(p)=0$ is left to the reader. Then, we have

$G=G(p)+G^{o}(O)(T-p),\text{ as functions on }O,$
$F=F({}^{G(p)}p)+F^{o}(O)(T-{}^{G(p)}p)\text{ as functions on }O.$

It easy to check that the following identity holds on $U$:

$F\diamond
G=F({}^{G(p)}p)G(p)+\left(F^{o}(O)G(p)+F\diamond(G^{o}(O))\right)\diamond(T-p).$

By Proposition 3.2, we have

$\displaystyle(F\diamond G)^{o}(p)=$
$\displaystyle\left(F^{o}(O)G(p)+F\diamond(G^{o}(O))\right)(\overline{p})$
$\displaystyle=$ $\displaystyle(F^{o}\diamond G)(p)+(F\diamond
G^{o})(\overline{p}).$

∎

Now, we give some examples of skew-convex functions on $\mathbb{H}$. The
functions $q\mapsto\text{Re}(q)^{n}$ and $q\mapsto|q|^{n}$, where $n$ is a
natural number, are skew convex on $\mathbb{H}$ since they are constant on all
orbits. As another examples of skew-convex functions, we have polynomials
$P(T)\in\mathbb{H}[T]$, where $\mathbb{H}[T]$ is the ring of polynomials in a
central indeterminate $T$ over $\mathbb{H}$ (see Example 2.2). Recall that a
(left) polynomial $P(T)=\sum_{m=0}^{n}q_{m}T^{m}\in\mathbb{H}[T]$, as a
function on $\mathbb{H}$, is evaluated as follows

$P(q)=\sum_{m=0}^{n}q_{m}q^{m}.$

It is easy to verify that the orbital derivative of
$P(T)=\sum_{m=0}^{n}q_{m}T^{m}\in\mathbb{H}[T]$ at an orbit $O(q)$ equals

$P^{o}(q)=\sum_{m=1}^{n}q_{m}\left(\sum_{k=0}^{m-1}\overline{q}^{k}q^{m-1-k}\right).$

To introduce the next class of skew-convex functions on $\mathbb{H}$, we need
the following result.

######  Proposition 3.4.

Let $X\subset\mathbb{H}$ be invariant and $F_{1},F_{2},...$ be a sequence of
skew-convex $\mathbb{H}$-valued functions on $X$. Suppose that the sequence
$F_{1},F_{2},...$ converges pointwise to a function $F\colon X\to\mathbb{H}$.
Then, $F$ is skew convex on $X$.

###### Proof.

One can easily prove the proposition using the definition of a skew-convex
function (see Definition 2.2). ∎

As an application of this proposition, we show that (left) spherical series
are skew convex. Let us review some facts regarding spherical series over
$\mathbb{H}$ (see also [11]). Suppose that $O\subset\mathbb{H}$ is an orbit.
We associate a polynomial $P_{O}(T)\in\mathbb{H}[T]$ to $O$ as follows: If
$O=\\{r\\}$ is a trivial orbit, we set $P_{O}(T)=T-r$. If $O$ is a nontrivial
orbit, we put

$P_{O}(T)=T^{2}-2\text{Re}(p)T+|p|^{2},\text{ where }p\in O.$

Note that $P_{O}(T)$ does not depend on the choice $p\in O$. By a (formal)
spherical series centered at $O$, we shall mean a series of the form

$S(T)=\sum_{n=0}^{\infty}q_{n}P_{O}(T)^{n},$

where $q_{0},q_{1},...\in\mathbb{H}$. The value $S(q)$ at a point
$q\in\mathbb{H}$ is defined to be

$S(q)=\sum_{n=0}^{\infty}q_{n}P_{O}(q)^{n},$

if the series converges. The notion of right spherical series is defined in a
similar fashion. We note that a spherical series centered at a trivial orbit
$O=\\{r\\}$ is just an ordinary series of the form

$\sum_{n=0}^{\infty}q_{n}(T-r)^{n}.$

The following proposition is easily proved using the Weierstrass $M$-test.

######  Proposition 3.5.

Let $\sum_{n=0}^{\infty}q_{n}P_{O}(T)^{n}$ be a spherical power series
centered at an orbit $O$. Suppose, furthermore, that
$\limsup_{n\to\infty}|q_{n}|^{1/n}=1/R$ for some $R>0$. Then, the series
converges absolutely and uniformly on the compact subsets of the region

$U(O,R)\colonequals\\{q\in\mathbb{H}\,|\,|P_{O}(q)|<R\\}.$

The number $R$ in this Proposition will be called the radius of convergence of
the spherical power series. The region $U(O,R)$ will be called the region of
convergence of the spherical power series. It follows immediately from
Proposition 3.4 that any spherical series which is convergent on an invariant
set $X$ is skew convex on $X$. It is clear that the orbital derivative of a
spherical series is zero at its center.

We conclude this part with the following lemma the proof of which is left to
the reader.

######  Lemma 3.6.

Let $F\colon U\to\mathbb{H}$ be a function on a region and $O\subset U$ be an
orbit. If $F$ is skew convex on $U\setminus O$ and is continuous at all $p\in
O$, then $F$ is skew convex on $U$.

###  3.2 Skew-invertible functions on $\mathbb{H}$

In this section, we study the question of skew invertibility for skew-convex
functions on invariant subsets of $\mathbb{H}$. We begin with the following
result.

######  Proposition 3.7.

Let $F\colon O\to\mathbb{H}$ be a skew-convex function on an orbit
$O\subset\mathbb{H}$. Then, $F$ is skew invertible if and only if $F(x)=0$ has
no solutions on $O$.

###### Proof.

The result is clear if $O$ is a trivial orbit. So, we assume that $O$ is
nontrivial. The “only if” direction follows from Part 3 of Lemma 2.8. We prove
the other direction. Since $F$ is skew-convex, there exists constants
$a,b\in\mathbb{H}$ such that $F(p)=a+bp$ for all $p\in O$. By Lemma 2.8 and
Proposition 2.10, we need only show that for any $p\in O$, there exists
$x\in\mathbb{H}^{*}$ such that $F({}^{x}p)=x^{-1}$. For a fixed $p\in O$, we
need to show that the equation

$a+b{}^{x}p=x^{-1}\text{ or }ax+bxp=1,$

has a nonzero solution. The assignment $x\mapsto ax+bxp$ defines an
$\mathbb{R}$-linear map on $\mathbb{H}$. Since its kernel is zero, it must be
onto, and therefore, there exists $x\in\mathbb{H}^{*}$ such that $ax+bxp=1$. ∎

As an example, we see that $T-q\in\mathbb{H}[T]$ is skew invertible on
$\mathbb{H}\setminus O(q)$ (see also Example 2.3). Moreover, we have the
following description of $(T-q)^{\langle-1\rangle}$.

######  Proposition 3.8.

For any $q_{0}\in\mathbb{H}$, the function $T-q_{0}$ is skew invertible on
$\mathbb{H}\setminus O(q_{0})$ and its skew inverse satisfies

$(T-q_{0})^{\langle-1\rangle}(q)=(q-\overline{q}_{0})\left(q^{2}-2\text{Re}(q_{0})q+|q_{0}|^{2}\right)^{-1},\text{
for all }q\notin O(q_{0}).$

In particular, we have

$(T-q_{0})^{\langle-1\rangle}(q)=\left(q-q_{0}\right)^{-1},\text{ for all
}q\notin O(q_{0})\text{ with }qq_{0}=q_{0}q.$

Moreover, $(T-q_{0})^{\langle-1\rangle}\colon\mathbb{H}\setminus
O(q_{0})\to\mathbb{H}^{*}$ is 1-1 and its image is the set

$\mathbb{H}^{*}\setminus\\{x\in\mathbb{H}\,|\,\text{Re}(x)=0\text{ and
}\text{Re}(q_{0}x)=-\frac{1}{2}\\}.$

###### Proof.

It is known that the equation $xq-q_{0}x=1$ has a (unique) solution
$x\in\mathbb{H}$ if $q\notin O=O(q_{0})$ (see [9]). In fact, the unique
solution of $xq-q_{0}x=1$ is

$x=(q-\overline{q}_{0})\left(q^{2}-2\text{Re}(q_{0})q+|q_{0}|^{2}\right)^{-1}.$

As seen in Example 2.3, it follows that $T-q_{0}$ is skew invertible on
$\mathbb{H}\setminus O$ and its skew inverse $(T-q_{0})^{\langle-1\rangle}$ is
skew convex. Moreover, we have

$(T-q_{0})^{\langle-1\rangle}(q)=(q-\overline{q}_{0})\left(q^{2}-2\text{Re}(q_{0})q+|q_{0}|^{2}\right)^{-1},$

for all $q\in\mathbb{H}\setminus O$. The fact that the skew inverse of
$T-q_{0}$ is 1-1 follows from proposition 2.9. To prove the last assertion, we
note that the equation $xq-q_{0}x=1$ gives $q=x^{-1}q_{0}x+x^{-1}$. It follows
from the above discussion that the image of $(T-q_{0})^{\langle-1\rangle}$
consists of all $x\in\mathbb{H}^{*}$ such that $x^{-1}q_{0}x+x^{-1}\notin O$.
It is easy to show that

$x^{-1}q_{0}x+x^{-1}\in O\iff\text{Re}(x)=0\text{ and
}\text{Re}(q_{0}x)=-\frac{1}{2},$

from which the last part of the proposition follows. ∎

######  Remark 3.1.

Similarly, one can show that $T-q_{0}$ is skew invertible on
$\mathbb{H}\setminus O(q_{0})$ with respect to the right skew product.
Moreover, the skew inverse $(T-q_{0})^{\langle-1\rangle_{r}}$ of $T-q_{0}$
satisfies

$(T-q_{0})^{\langle-1\rangle_{r}}(q)=\left(q^{2}-2\text{Re}(q_{0})q+|q_{0}|^{2}\right)^{-1}(q-\overline{q}_{0}),\text{
for all }q\notin O(q_{0}).$

In [2], the function $(T-q_{0})^{\langle-1\rangle_{r}}$ is studied in more
detail.

Now, we treat polynomials of higher degree. Let $P(T)\in\mathbb{H}[T]$ be a
nonzero polynomial. Set

$Z(P)=\\{q\in\mathbb{H}\,|\,\exists p\in O(q)\text{ such that }P(p)=0\\}.$

It follows directly from Proposition 3.7 that $P$ is skew invertible on the
set $\mathbb{H}\setminus Z(P)$. If $P(T)\in\mathbb{R}[T]$ is a polynomial with
real coefficients, then it is easy to see that

$P(T)^{\langle-1\rangle}(q)=P(q)^{-1},\text{ for all }q\notin Z(P).$

Therefore, there is no risk of confusion in using the notation $P(T)^{-1}$
instead of $P(T)^{\langle-1\rangle}$ when $P(T)$ has real coefficients. For a
detailed account of polynomials over quaternions, see [4, Chapter 3].

###  3.3 Limits and the skew product

In this part, we investigate the behavior of the skew product with respect to
limits. We begin with a simple lemma whose proof is left to the reader.

######  Lemma 3.9.

Let $U\subset\mathbb{H}$ be a region and $F,G\colon U\to\mathbb{H}$ be
functions. Suppose that the limit $p=\lim_{x\to q}G(x)$ exists for some
$q\in\mathbb{H}$. Then:
(1) If $p\neq 0$ and $\lim_{x\to{}^{p}q}F(x)$ exists, the limit $\lim_{x\to
q}(F\diamond G)(x)$ exists and

$\lim_{x\to q}(F\diamond G)(x)=(\lim_{x\to{}^{p}q}\,F(x))\lim_{x\to q}G(x).$

(2) If $p=0$ and $F$ is bounded on an open neighborhood of the orbit $O(q)$,
then $\lim_{x\to q}(F\diamond G)(x)=0$.

Next, we show that the skew product of continuous functions is continuous.

######  Proposition 3.10.

Let $F,G\colon U\to\mathbb{H}$ be functions on a region. Then:
(a) If $F$ and $G$ are continuous at each point of an orbit $O\subset U$, then
so is $F\diamond G$.
(b) If $F$ is skew invertible and continuous on $U$, then its skew inverse is
continuous on $U$.

###### Proof.

(a) It follows from Lemma 3.9 and the fact that any orbit in $\mathbb{H}$ is
compact.
(b) Let $F^{\langle-1\rangle}$ denote the skew inverse of $F$. By Lemma 2.8,
$F$ is nonzero on $U$ and the map $g(x)\colon={}^{F(x)}x$ is a bijection of
$U$ onto itself. Note that $g\colon U\to U$ is continuous. By Brouwer’s
invariance of domain theorem, $g$ is a homeomrphism, that is, its inverse
(with respect to composition) $h\colon U\to U$ is continuous. Since

$F^{\langle-1\rangle}(x)=F(h(x))^{-1},\text{ for all }x\in U,$

we conclude that $F^{\langle-1\rangle}$ is continuous. ∎

Regarding limits of skew products of sequences of functions, we have the
following result.

######  Proposition 3.11.

Let $\\{F_{n}\\}$ and $\\{G_{n}\\}$ be sequences of continuous functions on an
invariant compact subset $X$ of $\mathbb{H}$. Suppose that the sequences
$\\{F_{n}\\}$ and $\\{G_{n}\\}$ converge uniformly to functions $F,G\colon
X\to\mathbb{H}$, respectively. If the function $F$ is skew convex on $X$, then
the sequence $\\{F_{n}\diamond G_{n}\\}$ converges uniformly to the function
$F\diamond G$.

###### Proof.

Since $F$ is skew-convex, we can write

$F\diamond G-F_{n}\diamond G_{n}=(F_{n}-F)\diamond G_{n}+F\diamond(G_{n}-G).$

Note that $(F_{n}-F)\diamond G_{n}$ and $F\diamond(G_{n}-G)$ converge
uniformly to the zero function, from which the desired result follows. ∎

###  3.4 The skew derivative

We begin by introducing the definition of a skew-differentiable function.

######  Definition 3.2.

A function $F\colon U\to\mathbb{H}$ on a region $U\subset\mathbb{H}$ is said
to be (left) skew differentiable at $q\in U$ if there exists a function
$\mathcal{D}_{q}F\colon U\to\mathbb{H}$ which is continuous at all $p\in O(q)$
and satisfies

$F=F(q)+\mathcal{D}_{q}(F)\diamond(T-q),\text{ as functions on }U.$

A function $F\colon U\to\mathbb{H}$ is called (left) skew regular on a region
$U$ if it is (left) skew differentiable at all points in $U$.

Note that the function $\mathcal{D}_{q}(F)$ introduced in this definition is
unique because it is continuous on $O(q)$ and

$\mathcal{D}_{q}(F)=(F-F(q))\diamond(T-q)^{\langle-1\rangle},\text{ on
}U\setminus O(q).$

######  Remark 3.2.

The function $\mathcal{D}_{q}(F)$ has also been used in the context of slice-
regular functions. For more details, see [11, Section 2].

######  Remark 3.3.

Similarly, one can define the concept of right skew-differentiability using
the right skew product. We leave to the reader the analogous definitions and
results concerning right skew-differentiability. Since we mainly work with the
concept of left skew-differentiability, we shall often drop the adjective
“left”.

As a direct consequence of Proposition 3.2, we obtain the following result.

######  Proposition 3.12.

Let $F\colon U\to\mathbb{H}$ be a function on a region. If $F$ is skew
differentiable at some point of an orbit $O\subset U$, then $F$ is skew convex
on the orbit $O$. In particular, if $F$ is skew regular on $U$, then it is
skew convex on $U$.

We show that skew regularity implies continuity.

######  Proposition 3.13.

Let $F\colon U\to\mathbb{H}$ be a function on a region. If $F$ is skew
differentiable at $q\in U$, then $F$ is continuous at $q$.

###### Proof.

Since the orbit $O(q)$ is compact and $\mathcal{D}_{q}(F)$ is continuous at
all $p\in O(q)$, we see that $\mathcal{D}_{q}(F)$ is bounded on an open
neighborhood of $O(q)$. Now, the result follows from Part (2) of Lemma 3.9.

∎

Our definition of the (left) skew derivative is as follows.

######  Definition 3.3.

Let $F\colon U\to\mathbb{H}$ be a function on a region $U\subset\mathbb{H}$.
Suppose that $F$ is skew differentiable at $q_{0}\in U$. Then, the (left) skew
derivative of $F$, denoted by $F^{\prime}(q_{0})$, is defined to be the limit

$F^{\prime}(q_{0})=\lim_{O(q_{0})\notniFromTxfonts q\to
q_{0}}\left((F-F(q_{0}))\diamond(T-q_{0})^{\langle-1\rangle}\right)\,(q).$

In the particular case where $r\in U$ is real, we have

$F^{\prime}(r)=\lim_{q\to r}\left(F\left(q\right)-F(r)\right)\,(q-r)^{-1}.$

In order to derive some basic properties of the skew derivative, we need the
following lemma.

######  Lemma 3.14.

Let a function $F\colon U\to\mathbb{H}$ on a region $U\subset\mathbb{H}$ be
skew differentiable at $q\in U$. If $F$ is skew convex on $U$, so is
$\mathcal{D}_{q}(F)$, and we have

$\mathcal{D}_{q}(F)(p)=F^{\prime}(q)+\left(F^{\prime}(q)-F^{o}(q)\right)(q-\overline{q})^{-1}(p-q),\text{
for all }p\in O(q).$

###### Proof.

Note that the function $(T-q)^{\langle-1\rangle}$ is defined (by Proposition
3.8) and continuous on $U\setminus O(q)$ (by Proposition 3.10). In the light
of Lemma 3.6, the first part follows from the facts that
$\mathcal{D}_{q}(F)=\left(F-F(q)\right)\diamond(T-q)^{\langle-1\rangle}$ is
skew convex on $U\setminus O(q)$, and $\mathcal{D}_{q}(F)$ is continuous at
all $p\in O(q)$. The second part is a consequence of the facts that
$\mathcal{D}_{q}(F)$ is skew convex and
$\mathcal{D}_{q}(F)(q)=F^{\prime}(q),\,\mathcal{D}_{q}(F)(\overline{q})=F^{o}(q)$.
∎

In the following proposition, some further properties of the skew derivative
are summarized.

######  Proposition 3.15.

Let $F,G\colon U\to\mathbb{H}$ be functions on a region $U\subset\mathbb{H}$.
(1) Let $q\in U$. If $F$ and $G$ are skew differentiable at $q$, then $F+G$ is
also skew differentiable at $q$ and
$(F+G)^{\prime}(q)=F^{\prime}(q)+G^{\prime}(q)$.
(2) If $F,G$ are skew regular on $U$, then so is $F\diamond G$. Moreover, we
have

$(F\diamond G)^{\prime}=F^{\prime}\diamond G+F\diamond G^{\prime}.$

(3) If $F$ is skew regular and skew invertible on $U$, then its skew inverse
is skew regular on $U$, and moreover, we have

$\left(F^{\langle-1\rangle}\right)^{\prime}=-F^{\langle-1\rangle}\diamond
F^{\prime}\diamond F^{\langle-1\rangle}.$

###### Proof.

(1) Trivial.
(2) Fix $q\in U$. We only treat the case $G(q)\neq 0$, and leave the easier
case $G(q)=0$ to the reader. Set

$E_{1}=\mathcal{D}_{{}^{G(q)}q}(F)\text{ and }E_{2}=\mathcal{D}_{q}(G).$

We have

$F=F({}^{G(q)}q)+E_{1}\diamond(T-{}^{G(q)}q)\text{ and
}G=G(q)+E_{2}\diamond(T-q),\text{ on }U.$

Since $F$ and $E_{1}$ are skew convex (see Proposition 3.12 and Lemma 3.14),
we have

$F\diamond G=(F\diamond G)(q)+\left(E_{1}\diamond G(q)+F\diamond
E_{2}\right)\diamond(T-q).$

By Lemma 3.9, the function $E_{1}\diamond G(q)+F\diamond E_{2}$ is continuous
at all $p\in O(q)$. Therefore, $F\diamond G$ is skew regular at $q$ and

$\mathcal{D}_{q}(F\diamond G)=\mathcal{D}_{{}^{G(q)}q}(F)\diamond
G(q)+F\diamond\mathcal{D}_{q}(G).$

Evaluating at $q$, we obtain

$(F\diamond G)^{\prime}(q)=(F^{\prime}\diamond G)(q)+(F\diamond
G^{\prime})(q).$

(3) Fix $p\in U$ and set $q={}^{F^{\langle-1\rangle}(p)}p$. Note that the
quaternion $F(q)$ is the inverse of the quaternion $F^{\langle-1\rangle}(p)$.
Set $E=\mathcal{D}_{q}(F).$ We can write

$\displaystyle F^{\langle-1\rangle}-F^{\langle-1\rangle}(p)=$
$\displaystyle-F^{\langle-1\rangle}\diamond(F-F(q))\diamond
F^{\langle-1\rangle}(p)$ $\displaystyle=$
$\displaystyle-F^{\langle-1\rangle}\diamond E\diamond(T-q)\diamond
F^{\langle-1\rangle}(p)$ $\displaystyle=$
$\displaystyle-F^{\langle-1\rangle}\diamond E\diamond
F^{\langle-1\rangle}(p)\diamond(T-p),$

from which the result follows.

∎

As an application of this proposition, we show that the skew derivative is
skew convex.

######  Corollary 3.16.

Let $F\colon U\to\mathbb{H}$ be a function on a region. If $F$ is skew regular
on $U$, then $F^{\prime}$ is skew convex on $U$.

###### Proof.

Using Proposition 3.12, we see that $F$ is skew convex on $U$. By definition,
we have

$F\diamond(a+b)=F\diamond a+F\diamond b,\text{ for all }a,b\in\mathbb{H}.$

Part (2) of Proposition 3.15 shows that

$F^{\prime}\diamond(a+b)=F^{\prime}\diamond a+F^{\prime}\diamond b,\text{ for
all }a,b\in\mathbb{H}.$

Therefore, $F^{\prime}$ is skew convex on $U$. ∎

Next, we present the chain rule for skew-differentiable functions.

######  Proposition 3.17.

Let $\phi\colon U\to U^{\prime}$ be a continuous action-preserving function
between regions in $\mathbb{H}$. If $\phi$ is skew differentiable at some
$q_{0}\in U$ and a function $F\colon U^{\prime}\to\mathbb{H}$ is skew
differentiable at $\phi(q_{0})$, then $F\circ\phi$ is skew differentiable at
$q_{0}$. Moreover, we have

$\mathcal{D}_{q_{0}}(F\circ\phi)=\left(\mathcal{D}_{\phi(q_{0})}(F)\circ\phi\right)\diamond\mathcal{D}_{q_{0}}(\phi).$

In particular,
$(F\circ\phi)^{\prime}(q_{0})=F^{\prime}(\phi(q_{0}))\phi^{\prime}(q_{0})$ if
$\phi^{\prime}(q_{0})$ commutes with $\phi(q_{0})$. We also have
$(F\circ\phi)^{o}(q_{0})=F^{o}(\phi(q_{0}))\phi^{o}(q_{0})$.

###### Proof.

Set

$E_{1}=\mathcal{D}_{\phi(q_{0})}(F)\text{ and
}E_{2}=\mathcal{D}_{q_{0}}(\phi).$

Then

$F=F(\phi(q_{0}))+E_{1}\diamond(T-\phi(q_{0}))\text{ on }U^{\prime},$
$\phi=\phi(q_{0})+E_{2}\diamond(T-q_{0})\text{ on }U.$

Using the fact that $\phi$ is action preserving, one can easily check that

$F\circ\phi=(F\circ\phi)(q_{0})+\left((E_{1}\circ\phi)\diamond
E_{2}\right)\diamond(T-q_{0})\text{ on }U.$

Since $(E_{1}\circ\phi)\diamond E_{2}$ is continuous on $U$, we conclude that
$F\circ\phi$ is skew differentiable at $q_{0}$, and

$\mathcal{D}_{q_{0}}(F\circ\phi)=\left(\mathcal{D}_{\phi(q_{0})}(F)\circ\phi\right)\diamond\mathcal{D}_{q_{0}}(\phi).$

The chain rule for the orbital derivative is trivial. ∎

Regarding the inverse of a skew-differentiable function with respect to
function composition, we prove the following result.

######  Proposition 3.18.

Let $\phi\colon U\to\mathbb{H}$ be a continuous action-preserving function on
a region $U$. Suppose that $\psi:U^{\prime}\to\mathbb{H}$ is a continuous
function on a region $U^{\prime}$ such that $\phi(U)\subset U^{\prime}$ and
$\psi(\phi(q))=q$ for all $q\in U$. If $\psi$ is skew differentiable at some
$\phi(q_{0})\in U^{\prime}$ and the function $\mathcal{D}_{\phi(q_{0})}(\psi)$
does not have a root on the orbit $O(\phi(q_{0}))$, then $\phi$ is skew
differentiable at $q_{0}$. Moreover, we have

$\mathcal{D}_{q_{0}}(\phi)=(\mathcal{D}_{\phi(q_{0})}(\psi)\circ\phi)^{\langle-1\rangle}.$

In particular, $\phi^{\prime}(q_{0})=\psi^{\prime}(\phi(q_{0}))^{-1}$ if
$\phi^{\prime}(q_{0})$ commutes with $\phi(q_{0})$.

###### Proof.

Setting $E=\mathcal{D}_{\phi(q_{0})}(\psi)$, we have

$\psi=q_{0}+E\diamond(T-\phi(q_{0}))\text{ on }U^{\prime}.$

Composing with $\phi$ and using the fact that $\phi$ is action-preserving, we
obtain

$T=q_{0}+(E\circ\phi)\diamond(\phi-\phi(q_{0}))\text{ on }U.$

Since $\phi$ is 1-1 and $E\circ\phi$ does not have a zero on the orbit
$O(q_{0})$, one can see that $E\circ\phi$ is nowhere zero on $U$. By
Proposition 3.7, $E\circ\phi$ is skew invertible on $U$. It follows that

$\phi=\phi(q_{0})+(E\circ\phi)^{\langle-1\rangle}\diamond(T-q_{0})\text{ on
}U.$

By Part (b) of Proposition 3.10, $(E\circ\phi)^{\langle-1\rangle}$ is
continuous. Therefore, $\phi$ is skew differentiable at $q_{0}$, and

$\mathcal{D}_{q_{0}}(\phi)=(\mathcal{D}_{\phi(q_{0})}(\psi)\circ\phi)^{\langle-1\rangle}.$

∎

As an application of this proposition, one can show that the principal
quaternionic logarithm $Log:\mathbb{H}\setminus(-\infty,0]\to\mathbb{H}$ (see
[7, Definition 3.4]) is skew regular, using the facts that $e^{Log(q)}=q$, for
all $q\in\mathbb{H}\setminus(-\infty,0]$, and $e^{q}$ is a continuous action-
preserving function on $\mathbb{H}$. Moreover, we have
$Log^{\prime}(q)=q^{-1}$ for all $q\in\mathbb{H}\setminus(-\infty,0]$. The
details are left to the reader.

We conclude this part with some remarks.

######  Remark 3.4.

In [3], the regular product of slice-regular functions is first defined for
the series centered at $0$, and then extended to slice-regular functions (see
[3, Definition 5.3]). Proposition 5.12 in the same reference gives a formula
for the regular product of regular functions over axially symmetric slice
domains. This formula is just the formula for the right skew product (see
Definition 2.2).

######  Remark 3.5.

We will prove that any slice-regular function over a symmetric slice domain is
skew regular (see Theorem 3.20). It follows from Proposition 3.12 that slice-
regular functions over symmetric slice domains are skew-convex. The so-called
representation formulas (see for example [5, Formulas 1.8,1.9,1.10]) are
simply consequences of the fact that slice-regular functions over symmetric
slice domains are skew-convex.

######  Remark 3.6.

The notion of orbital derivative, defined in Definition 3.1, is also
introduced in [11], where it is called spherical derivative. In [8], a
Leibniz-type formula is given for the orbital product. The formula given in
Proposition 3.3 is a new formula.

######  Remark 3.7.

In [4, Proposition 1.28], it is shown that slice-regular functions over a
symmetric slice domain form a noncommutative ring under the regular product.
The Leibniz formula for the slice derivative is also given in the same
reference (see [4, Proposition 1.40]). Proposition 3.15 gives the
corresponding results for skew-regular functions.

######  Remark 3.8.

The author has not been able to find any form of the chain rule for slice-
regular functions in the literature. The only result in this direction seems
to be Lemma 1.32 in [5] which, roughly speaking, states that the composition
of two slice-regular functions is slice-regular if one of them is slice
preserving.

###  3.5 Skew regularity and slice regularity

In this section, we show that skew-regular functions are slice regular, and
moreover, the converse holds over symmetric slice domains. We need to review
some facts regarding slice-regular functions. For more details, see [5] and
references therein. Let $\mathbb{S}$ denote the set of all $I\in\mathbb{H}$
such that $I^{2}=-1$. For $I\in\mathbb{S}$ and a subset $A$ of $\mathbb{H}$,
put $A_{I}=A\cap(\mathbb{R}+\mathbb{R}I)$. Note that $\mathbb{R}+\mathbb{R}I$
is a commutative subfield of $\mathbb{H}$. In fact, it is isomorphic to
$\mathbb{C}$ via $x+yI\mapsto x+yi$, using which we shall identify
$\mathbb{R}+\mathbb{R}I$ with the complex plane $\mathbb{C}$.

######  Definition 3.4.

A function $F\colon U\to\mathbb{H}$ on an open set $U\subset\mathbb{H}$ is
called (right) slice regular if for each $I\in\mathbb{S}$, the restriction
$F_{I}$ of $F$ to $U_{I}$ is holomorphic in the sense that it has continuous
partial derivatives and

$\frac{1}{2}\left(\frac{\partial F_{I}}{\partial x}+\frac{\partial
F_{I}}{\partial y}\,I\right)=0,\text{ on }U_{I}.$

If $F$ is slice regular on $U$, its slice derivative $F_{s}^{\prime}(q)$ at
$q\in U_{I}$ is defined to be

$F_{s}^{\prime}(q)=\frac{1}{2}\left(\frac{\partial F_{I}}{\partial
x}-\frac{\partial F_{I}}{\partial y}\,I\right).$

First, we show that every skew-differentiable function is slice regular.

######  Proposition 3.19.

Let $F\colon U\to\mathbb{H}$ be a function on a region. If $F$ is skew regular
on $U$, then it is slice regular on $U$, and moreover, its skew derivative is
equal to its slice derivative.

###### Proof.

Fix $I\in\mathbb{S}$. We need to show that the restriction of $F$ to the set
$U_{I}$ satisfies the Cauchy–Riemann equations. We choose $J\in\mathbb{S}$
such that $J$ is orthogonal to $I$, that is, $IJ=-JI$. Set $z=x+yI$, where
$x,y\in\mathbb{R}$. There exist functions $F_{1},F_{2}\colon
U_{I}\to\mathbb{R}+\mathbb{R}I$ such that

$F(z)=F_{1}(z)+JF_{2}(z),\text{ for all }z\in U_{I}.$

Let $z_{0}=x_{0}+y_{0}I\in U_{I}$ be arbitrary. Since $F$ is skew
differentiable at $z_{0}$, we have

$F=F(z_{0})+\mathcal{D}_{z_{0}}(F)\diamond(T-z_{0}),\text{ as functions on
}U.$

The fact that $\mathbb{R}+\mathbb{R}I$ is a commutative field implies that

$F(z)=F(z_{0})+\mathcal{D}_{z_{0}}(F)(z)(z-z_{0}),\text{ for all }z\in U_{I},$

from which we obtain

$F_{1}^{\prime}(z_{0})+JF_{2}^{\prime}(z_{0})=\mathcal{D}_{z_{0}}(F)(z_{0})=F^{\prime}(z_{0}).$

In particular, $F_{1}$ and $F_{2}$ are holomorphic on $U_{I}$. Therefore, $F$
is slice regular. The identity $F^{\prime}=F^{\prime}_{s}$ can easily be
verified. ∎

In general, a slice-regular function may not be skew regular. In fact, there
exist slice-regular functions which are not even continuous (see Example 1.10
in [5]). To exclude such pathologies, one is forced to restrict one’s
attention to a certain class of open sets known as slice domains. Recall that
a connected open subset $U$ of $\mathbb{H}$ is called a slice domain if
$U\cap\mathbb{R}$ is nonempty and the intersection
$U\cap(\mathbb{R}+\mathbb{R}q)$ is connected for all $q\in
U\setminus\mathbb{R}$. It is known that any slice-regular function on a slice
domain $U$ can uniquely be extended to a slice-regular function on the
symmetric completion of $U$ (see [5, Theorem 1.24]). We remark that the
symmetric completion of a set $A\subset\mathbb{H}$ is the smallest invariant
subset of $\mathbb{H}$ which contains $A$. It turns out that the concepts of
slice regularity and skew regularity coincide over symmetric (meaning
invariant) slice domains.

######  Theorem 3.20.

Let $U\subset\mathbb{H}$ be a symmetric slice domain. A function $F\colon
U\to\mathbb{H}$ is slice regular on $U$ if and only if it is skew regular on
$U$, in which case its skew derivative equals its slice derivative.

###### Proof.

The “if” direction has been proved in Proposition 3.19. Conversely, let the
function $F\colon U\to\mathbb{H}$ be slice regular. Fix $q\in U$. It follows
from Theorem 3.17 in [5] that there exists a slice-regular function $E\colon
U\to\mathbb{H}$ such that

$F-F(q)=E\diamond(T-q),\text{ as functions on }U.$

Since $E$ is slice regular on a symmetric slice domain, $E$ must be continuous
on $U$. Therefore, $F$ is skew differentiable at $q$. As the slice derivative
is equal to $E(q)$ by the product rule for the slice derivative (see [5,
Proposition 1.40]), the proof is complete. ∎

Next, we show that strongly analytic functions are skew differentiable. Recall
that a function is called strongly analytic on a region if at every point $p$
in the region, the function can locally be represented as a convergent series
centered at $p$. For more details, see [4].

######  Proposition 3.21.

Let $F\colon U\to\mathbb{H}$ be strongly analytic on a region $U$. Then, $F$
is skew regular on $U$.

###### Proof.

Fix $p\in U$. Since $F$ is strongly analytic at $p$, we can find an open
neighborhood $p\in U_{0}\subset U$, and $q_{0},q_{1},...\in\mathbb{H}$ such
that

$F=\sum_{m=0}^{\infty}q_{m}(T-p)^{m},\text{ as functions on }U_{0},$

where the series converges absolutely and uniformly on $U_{0}$. Set

$E=\sum_{m=1}^{\infty}q_{m}(T-p)^{m-1}.$

Then $E$ is a continuous function on $U_{0}$ since the convergence is uniform.
Using Proposition 3.11, one can see that

$F=F(p)+E\diamond(T-p),\text{ as functions on }U_{0}.$

It follow that $F$ is skew differentiable at $p$. ∎

###  3.6 Spherical series representation of skew-regular functions

In this section, we show that skew-regular functions can locally be
represented by spherical series. Let us begin with the following result which
is an improvement of the fact that any spherical series is slice regular on
its region of convergence (see [11, Proposition 1.11]).

######  Proposition 3.22.

Let $S(T)=\sum_{n=0}^{\infty}q_{n}P_{O}(T)^{n}$ be a formal spherical power
series centered at an orbit $O$, and with radius of convergence $R>0$. Then,
$S(T)$ is skew regular on $U(O,R)=\\{q\in\mathbb{H}\,|\,|P_{O}(q)|<R\\}.$
Moreover, the skew derivative of $S$ is given by the formula

$S^{\prime}(T)=\sum_{n=1}^{\infty}q_{n}P_{O}(T)^{n-1}P^{\prime}_{O}(T),$

where the series is absolutely and uniformly convergent on the compact subsets
of $U(O,R)$.

###### Proof.

We only prove the first assertion. The proof of the second assertion is
similar to that of the corresponding result for ordinary power series over
real or complex numbers. If $O$ is a trivial orbit, the result follows from
Theorem 16 in [4] since every skew-regular function is slice regular by
Proposition 3.19. Assume that $O$ is a nontrivial orbit with
$P_{O}(T)=T^{2}-2x_{0}T+y_{0}^{2}$. Fix $q_{0}\in U(O,R)$. For every $n\geq
1$, there exists $P_{n}(T)\in\mathbb{H}[T]$ such that

$P_{O}(T)^{n}=P_{O}(q_{0})^{n}+P_{n}(T)(T-q_{0}),\text{ as elements of
}\mathbb{H}[T].$

It is easy to check that $P_{1}(T)=T+q_{0}-2x_{0}$. Using induction on $n$, we
obtain the following recursion formula

$P_{n+1}(T)=P_{O}(q_{0})^{n}(T+q_{0}-2x_{0})+P_{O}(T)P_{n}(T).$

Consider the series $E(T)=\sum_{n=0}^{\infty}q_{n}P_{n}(T)$. I claim that this
series is uniformly convergent on the compact subsets of $U(O,R)$. Let $q\in
U(O,R)$ and $m$ be the maximum of the norms $|P_{O}(q)|$ and $|P_{O}(q_{0})|$.
Using the recursion formula, one can easily show that

$|P_{n}(q)|\leq n|q+q_{0}-2x_{0}|m^{n-1},\text{ for all }n\geq 1.$

Now, the claim is easily proved using the Weierstrass $M$-test. In particular,
we see that $E$ is continuous on $U(O,R)$. Since

$S=S(q_{0})+E\diamond(T-q_{0}),$

the result follows. ∎

We need some preparations about line integrals over quaternions. Suppose that
a curve $\gamma\colon[a,b]\to\mathbb{H}$ is piecewise smooth. For
$\mathbb{H}$-valued functions $F,G$, defined and being continuous on the image
of $\gamma$, we will use the following line integral along $\gamma$:

$\int_{\gamma}F(q)\,dq\,G(q)\colonequals\int_{a}^{b}F(\gamma(t))\gamma^{\prime}(t)G(\gamma(t))dt.$

In the following proposition, we use the skew inverse of $T-p$ with respect to
the right skew product (see Remark 3.1).

######  Proposition 3.23.

Let $F\colon U\to\mathbb{H}$ be skew regular on a region and $O\subset U$ be a
nontrivial orbit such that $U$ contains the closure of the set

$U(O,R)=\\{q\in\mathbb{H}\,|\,|P_{O}(q)|<R\\},\text{ for some }R>0.$

For any $I\in\mathbb{S}$, we have

$F(p)=\frac{1}{2\pi}\int_{\gamma}F(q)\,(-Idq)\,(T-p)^{\langle-1\rangle_{r}}(q),\text{
for all }p\in U(O,R),$

where $\gamma$ is the boundary of the set
$U(O,R)\cap(\mathbb{R}+\mathbb{R}I)$. Moreover, we have

$F^{o}(p)=\frac{1}{2\pi}\int_{\gamma}F(q)\,(-Idq)\,P_{O(p)}(q)^{-1},\,\text{
for all }p\in U(O,R).$

###### Proof.

Fix $p^{\prime}\in U(O,R)$. Then,
$O(p^{\prime})\cap(\mathbb{R}+\mathbb{R}I)=\\{p_{0},\overline{p}_{0}\\}$ for
some $p_{0}\in\mathbb{H}$. An argument similar to the proof of Lemma 6.3 in
[5] shows that the desired formula holds for $p=p_{0}$ and
$p=\overline{p}_{0}$, since $(T-p)^{\langle-1\rangle_{r}}(q)=(q-p)^{-1}$ when
$pq=qp$. It is easy ot verify that the assignment

$p\mapsto\frac{1}{2\pi}\int_{\gamma}F(q)\,(-Idq)\,(T-p)^{\langle-1\rangle_{r}}(q),$

defines a (left) skew-convex function on $O(p^{\prime})$. Since $F$ is skew
convex on $O(p^{\prime})$, it follows immediately that the desired formula
holds on $O(p^{\prime})$. This completes the proof of the first formula. The
second formula is derived using the first formula and the definition of the
orbital derivative. ∎

Let us give two remarks about this propositions.

######  Remark 3.9.

The first formula given in Proposition 3.23 can be regarded as an analogue of
the Cauchy formula for holomorphic functions. Although our formula looks
similar to Formula 6.4 in [5], there is a fundamental difference between the
two formulas. Using our notations, we can rewrite the left-hand analogue of
Formula 6.4 in [5] as

$F(p)=\frac{1}{2\pi}\int_{\gamma}F(q)\,(-Idq)\,(q-T)^{\langle-1\rangle}(p).$

This formula does not seem suitable for deriving spherical series
presentations of skew-regular functions (see Proposition 3.25).

######  Remark 3.10.

As a consequence of Proposition 3.23, we see that the orbital derivative of a
skew-regular function is continuous. In fact, one can use the second formula
in the proposition to show that the orbital derivative of a skew-regular
function is real analytic. However, it may not be skew regular.

We also need the following lemma.

######  Lemma 3.24.

Let $p\in\mathbb{H}$, and $O\subset\mathbb{H}$ be a nontrivial orbit such that
$p\notin O$. Then, the right skew-regular function
$(T-p)^{\langle-1\rangle_{r}}$ can be represented as the (right) series

$(T-p)^{\langle-1\rangle_{r}}=\sum_{n=0}^{\infty}P_{O}(T)^{-n-1}(T+p-2x_{0})P_{O}(p)^{n},$

where the series is absolutely and uniformly convergent on the compact subsets
of the region

$\\{q\in\mathbb{H}|\,|P_{O}(p)|<|P_{O}(q)|\\}.$

###### Proof.

We have

$P_{O}(T)=P_{O}(p)+(T-p)\diamond_{r}(T+p-2x_{0}),\text{ as functions on
}\mathbb{H}.$

Note that $P_{O}(T)\in\mathbb{R}[T]$ is skew invertible on
$\mathbb{H}\setminus O(p)$ with respect to the left and right skew products
(see the discussion at the end of Section 3.2). It follows that

$1-P_{O}(p)\diamond_{r}P_{O}(T)^{-1}=(T-p)\diamond_{r}(T+p-2x_{0})\diamond_{r}P_{O}(T)^{-1},$

as functions on $\mathbb{H}\setminus O(p)$. Then, one can see that

$(T-p)^{\langle-1\rangle_{r}}=(T+p-2x_{0})\diamond_{r}P_{O}(T)^{-1}\diamond_{r}\left(1-P_{O}(p)\diamond_{r}P_{O}(T)^{-1}\right)^{\langle-1\rangle_{r}},$

as functions on $\\{q\in\mathbb{H}|\,|P_{O}(p)|<|P_{O}(q)|\\}.$ Now, the
result follows from the identity

$\left(1-P_{O}(p)\diamond_{r}P_{O}(T)^{-1}\right)^{\langle-1\rangle_{r}}=\sum_{n=0}^{\infty}P_{O}(T)^{-n}P_{O}(p)^{n},$

where the (right) series converges absolutely and uniformly on the compact
subsets of the region $\\{q\in\mathbb{H}|\,|P_{O}(p)|<|P_{O}(q)|\\}.$ ∎

Now, we can prove that skew-regular functions can locally be presented as
spherical series.

######  Proposition 3.25.

Let $F\colon U\to\mathbb{H}$ be skew regular on a region $U$. Let $O\subset U$
be a nontrivial orbit such that $U$ the closure of the set

$U(O,R)=\\{q\in\mathbb{H}\,|\,|P_{O}(q)|<R\\}.$

Then, there exist spherical series $S_{1}(T),S_{2}(T)$, centered at $O$, which
are convergent on $U(O,R)$ and satisfy

$F(T)=S_{1}(T)+S_{2}(T)\diamond T,\text{ as functions on }U(O,R).$

###### Proof.

Fix $I\in\mathbb{S}$ and let $\gamma$ be the boundary of
$U(O,R)\cap(\mathbb{R}+\mathbb{R}I)$. An application of Proposition 3.23 gives

$F(p)=\frac{1}{2\pi}\int_{\gamma}F(q)\,(-Idq)\,(T-p)^{\langle-1\rangle_{r}}(q),\text{
for all }p\in U(O,R).$

Note that $|P_{O}(p)|<|P_{O}(q)|=R$ for all $p\in U(O,R)$ and $q\in\gamma$.
Let $P_{O}(T)=T^{2}-2x_{0}T+y_{0}^{2}$. Using Lemma 3.24, we see that

$\displaystyle F(p)=$
$\displaystyle\frac{1}{2\pi}\int_{\gamma}F(q)\,(-Idq)\,(T-p)^{\langle-1\rangle_{r}}(q)$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{\gamma}F(q)\,(-Idq)\,\left(\sum_{n=0}^{\infty}P_{O}(q)^{-n-1}(q+p-2x_{0})P_{O}(p)^{n}\right)$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(\frac{1}{2\pi}\int_{\gamma}F(q)\,(-Idq)\,P_{O}(q)^{-n-1}(q+p-2x_{0})\right)P_{O}(p)^{n}.$

It is allowed to change the order of summation and integration since the
(right) series converges absolutely and uniformly on the compact set $\gamma$.
Setting

$S_{1}(T)=\sum_{n=0}^{\infty}\left(\frac{1}{2\pi}\int_{\gamma}F(q)\,(-Idq)\,P_{O}(q)^{-n-1}(q-2x_{0})\right)P_{O}(T)^{n},\text{
and }$
$S_{2}(T)=\sum_{n=0}^{\infty}\left(\frac{1}{2\pi}\int_{\gamma}F(q)\,(-Idq)\,P_{O}(q)^{-n-1}\right)P_{O}(T)^{n},$

we see that

$F(T)=S_{1}(T)+S_{2}(T)\diamond T,\text{ as functions on }U(O,R),$

completing the proof. ∎

Finally, we state a useful characterization of skew-regular functions.
Following [11, Definition 4.2], we first introduce a definition.

######  Definition 3.5.

A function $F\colon U\to\mathbb{H}$ on a region $U$ is called symmetrically
analytic if it has a spherical series representation at every orbit $O\subset
U$, that is,
(1) If $O\subset U$ is trivial, there exists a spherical series $S(T)$,
centered at $O$, which is convergent in a neighborhood $V\subset U$ of $O$ and
satisfy
$F(T)=S(T),\text{ as functions on }V.$
(2) If $O\subset U$ is nontrivial, there exist spherical series
$S_{1}(T),S_{2}(T)$, centered at $O$, which are convergent in a neighborhood
$V\subset U$ of $O$ and satisfy

$F(T)=S_{1}(T)+S_{2}(T)\diamond T,\text{ as functions on }V.$

Now we come to the main result of this section.

######  Theorem 3.26.

A function $F\colon U\to\mathbb{H}$ on a region $U\subset\mathbb{H}$ is skew
regular if and only if it is symmetrically analytic.

###### Proof.

The “if” direction follows from Proposition 3.22. The other direction follows
from Proposition 3.25 if $O$ is a nontrivial orbit. In the case of trivial
orbits, the result follows from Proposition 2.7 in [3]. ∎

Theorem 3.26 has a number of important consequences which will be studied in a
sequel to this paper. Let us just mention two direct consequences. The first
corollary is the generalization of the fact that the slice derivative of a
slice-regular function is slice regular.

######  Corollary 3.27.

If a function $F\colon U\to\mathbb{H}$ is skew regular on a region
$U\subset\mathbb{H}$, then it is infinitely skew differentiable. In
particular, the skew derivative $F^{\prime}\colon U\to\mathbb{H}$ of $F$ is
skew regular on $U$.

###### Proof.

This result follows from Theorem 3.26 and the fact that spherical series can
be differentiated term by term. The details are left to the reader. ∎

In the following corollary, $(T-p)^{\langle-n-1\rangle_{r}}$ denotes the
$(n+1)$-th power of $(T-p)^{\langle-1\rangle_{r}}$ with respect to the right
skew product.

######  Corollary 3.28.

Let $F\colon U\to\mathbb{H}$ be skew regular on a region and $O\subset U$ be a
nontrivial orbit such that $U$ contains the closure of the set

$U(O,R)=\\{q\in\mathbb{H}\,|\,|P_{O}(q)|<R\\}.$

For any $I\in\mathbb{S}$, we have

$F^{(n)}(p)=\frac{n!}{2\pi}\int_{\gamma}F(q)\,(-Idq)\,(T-p)^{\langle-n-1\rangle_{r}}(q),\text{
for all }p\in U(O,R),$

where $\gamma$ is the boundary of the set
$U(O,R)\cap(\mathbb{R}+\mathbb{R}I)$.

###### Proof.

In view of Corollary 3.27, one can use an argument similar to the proof of
Proposition 3.23 to derive the desired formula. ∎

## References

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# General Expressions for On-shell Recursion relations for Tree-level Open
String Amplitudes

Pongwit Srisangyingcharoen<EMAIL_ADDRESS>
###### Abstract

In this paper, we present a systematic derivation aimed at obtaining general
expressions for on-shell recursion relations for tree-level open string
amplitudes. Our approach involves applying the BCFW shift to an open string
amplitude written in terms of multiple Gaussian hypergeometric functions. By
employing binomial expansions, we demonstrate that the shifted amplitudes
manifest simple poles, which correspond to scattering channels of intermediate
states. Using the residue theorem, we thereby derive a general expression for
these relations.

###### keywords:

Bosonic string, string amplitudes, on-shell recursion relations

††journal: Physics letters B

organization=The Institute for Fundamental Study, Naresuan
University,city=Phisanulok, postcode=65000, country=Thailand

##  1 Introduction

Since the beginning of the string theory, understanding string scattering
amplitudes has been a fundamental concern among string theorists. It was long
known that in the limit of low energies, amplitudes in string theory reproduce
those in QFT such as Yang-Mills [1] and Einstein theory [2, 3] plus
$\alpha^{\prime}$ corrections [4, 5, 6, 7, 8]. The connection between string
theory and QFT provides useful applications in both theories. Understanding
the structure of string amplitudes would provide a better insight into those
of quantum field theories. Concrete examples are the celebrated Kawai-
Lewellen-Tye (KLT) relations [9] which relates closed string amplitudes in
terms of products of two open string amplitudes giving alternative
descriptions of gravity as the square of gauge theory. These non-linear
relations were proven later in the context of QFT [10, 11].

Another interesting structure was discovered by Plahte [12] which are linear
relations among color-ordered open string scattering amplitudes. These are
currently known as monodromy relations. In the field theory limit, the
relations reduce to the BCJ relations of Bern, Carrasco and Johansson [13] and
the Kleiss-Kujif relations [14]. This results in a reduction of the number of
color-ordered amplitudes from $(n-1)!$ as given by a cyclic property of the
trace down to $(n-3)!$ [15, 16]. The monodromy relations among partial open
string amplitudes can be captured by polygons in the complex plane [17].

During the early 2000s, advancements in the study of scattering amplitudes
were notably influenced by the discovery of the Britto-Cachazo-Feng-Witten
(BCFW) on-shell recursion relations [18, 19]. These relations enabled the
expression of tree-level amplitudes as products involving amplitudes with
fewer particles. The key idea for deriving the on-shell recursion relations is
based on the fact that any tree-level scattering amplitude is a rational
function of the external momenta, thus, one can turn an amplitude $A_{n}$ into
a complex meromorphic function $A_{n}(z)$ by deforming the external momenta
through introducing a complex variable $z$. These deformed momenta, satisfying
momentum conservation, are required to remain on-shell. For a scattering
process involving $n$ particles, the selection of an arbitrary pair of
particle momenta for shifting is permissible. Our choice is given by

$\displaystyle k_{1}\rightarrow\hat{k}_{1}(z)=$ $\displaystyle k_{1}-qz$ (1a)
$\displaystyle k_{n}\rightarrow\hat{k}_{n}(z)=$ $\displaystyle k_{n}+qz$ (1b)

where $q$ is a reference momentum which obeys $q\cdot q=k_{1}\cdot
q=k_{n}\cdot q=0$.

The unshifted amplitude $A_{n}(z=0)$ can be obtained from a contour
integration in which the contour is large enough to enclose all finite poles.
According to the Cauchy’s theorem,

$A_{n}(0)=\oint
dz\frac{A_{n}(z)}{z}-\sum_{\text{poles}}\text{Res}_{z=z_{\text{poles}}}\left(\frac{A_{n}(z)}{z}\right),$
(2)

the unshifted amplitude at $z=0$ is equal to the sum of the residues over all
the finite poles if the amplitude is well-behaved at large $z$ (which is the
case for most theories). For Yang-Mills theory, the residue at a finite pole
is the product of two fewer-point amplitudes with an on-shell exchanged
particle. In Yang-Mills a sum over the helicities of the intermediate gauge
boson and in general theories a sum over all allowed intermediate particle
states must also be done. In the general case, the BCFW recursion relation is

$A_{n}(0)=\sum_{\begin{subarray}{c}\text{poles}\\\
\alpha\end{subarray}}\sum_{\begin{subarray}{c}\text{physical}\\\
\text{states}\end{subarray}}A_{L}(\dots,P(z_{\alpha}))\frac{2}{P^{2}+M^{2}}A_{R}(-P(z_{\alpha}),\dots)$
(3)

with $P$ being the momentum of the exchanged particle with mass $M$.

The validity of equation (2) requires the absence of a pole at infinity. In
the case that there exists such a pole, one must include the residue at
infinity. However, the residue at this pole does not have a similar physical
interpretation to the residues at finite poles. A detailed discussion can be
found in [20].

This paper aims to present expressions for on-shell recursion relations
concerning open string amplitudes. While existing literature has explored
relations for open strings [21, 22, 23, 24], general forms for open string on-
shell recursion relations have never been delivered. Our paper aims to present
a systematic derivation to such relations based on the Koba-Nielsen integral
forms [25].

##  2 On-shell Recursion Relations for Four-point Tachyonic Open String
Amplitudes

Before deriving a general expression for the on-shell recursion relations of
open string amplitudes, let’s start with a discussion of the simplest example,
i.e. the four-point tachyon amplitude. Consider the partial amplitude

$\displaystyle\mathcal{A}(s,t)=\int_{0}^{1}dx\
x^{\alpha^{\prime}s-2}(1-x)^{\alpha^{\prime}t-2}$ (4)

where $s$ and $t$ are the Mandelstam variables given by $s=(k_{1}+k_{2})^{2}$
and $t=(k_{1}+k_{4})^{2}$. The open string vertex variables $x_{1},x_{3}$ and
$x_{4}$ are fixed to $0,1$ and $\infty$ due to the $PSL(2,R)$ gauge symmetry.
Under the shift in (1), $\mathcal{A}(s,t)\rightarrow\mathcal{A}(z^{\prime})$
giving

$\displaystyle\mathcal{A}(z^{\prime})=\int_{0}^{1}dx\
x^{\alpha^{\prime}s-2-z^{\prime}}(1-x)^{\alpha^{\prime}t-2}$ (5)

where $z^{\prime}=2\alpha^{\prime}zq\cdot k_{2}$. To determine poles generated
from $z^{\prime}$, we introduce the variable

$x=\exp{\left(-\frac{y}{\alpha^{\prime}s-2-z^{\prime}}\right)}\qquad\text{for}\quad\text{Re}(\alpha^{\prime}s-2-z^{\prime})>0.$
(6)

The integral (5) becomes

$\displaystyle\mathcal{A}(z^{\prime})=\sum_{a=0}^{\infty}\binom{\alpha^{\prime}t-2}{a}(-1)^{a}\int_{0}^{\infty}dy\
e^{-y}\frac{e^{-\frac{(1+a)y}{\alpha^{\prime}s-2-z^{\prime}}}}{\alpha^{\prime}s-2-z^{\prime}}$
(7)

where a binomial expansion
$(1-x)^{a}=\sum_{k=0}^{\infty}\binom{a}{k}(-1)^{k}x^{k}$ was applied.
$\binom{a}{k}$ denotes a binomial coefficient. Expanding a Taylor series for
the exponential, one obtains

$\displaystyle\mathcal{A}(z^{\prime})$
$\displaystyle=\sum_{a=0}^{\infty}\binom{\alpha^{\prime}t-2}{a}(-1)^{a}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\int_{0}^{\infty}dy\
e^{-y}\frac{((1+a)y)^{k}}{(\alpha^{\prime}s-2-z^{\prime})^{k+1}}.$ (8)
$\displaystyle=-\sum_{a=0}^{\infty}\binom{\alpha^{\prime}t-2}{a}(-1)^{a}\sum_{k=0}^{\infty}\frac{(1+a)^{k}}{(z^{\prime}-(\alpha^{\prime}s-2))^{k+1}}$
(9)

This shows that $\mathcal{A}(z^{\prime})$ contains the $k^{\text{th}}$-order
poles at $z^{\prime}=\tilde{z}\equiv\alpha^{\prime}s-2$. We can regain the
unshifted amplitude $\mathcal{A}(0)$ using the relation (2). Therefore,

$\displaystyle\mathcal{A}(0)$
$\displaystyle=-\text{Res}_{z^{\prime}=\tilde{z}}\left(\frac{\mathcal{A}(z^{\prime})}{z^{\prime}}\right)=\sum_{a,k=0}^{\infty}\binom{\alpha^{\prime}t-2}{a}\frac{(-1)^{a+k}}{\tilde{z}}\left(\frac{1+a}{\tilde{z}}\right)^{k}$
$\displaystyle=\sum_{a=0}^{\infty}\binom{\alpha^{\prime}t-2}{a}\frac{(-1)^{a}}{\alpha^{\prime}s-(1-a)}.$
(10)

The expression implies propagators of intermediate on-shell string states.
Comparing to the BCFW recursion relation (3), the residues of (10) are sum
over product of physical state amplitudes at each fixed level $a$. This was
explicit shown in [23].

It is worth noting that the derivation of the expression (10) assumes the
condition such that Re$(\alpha^{\prime}s-2-z^{\prime})>0$ during the
coordinates transformation (6). To obtain the same argument but for the
kinematic regime Re$(\alpha^{\prime}s-2-z^{\prime})<0$ is a bit trickier as we
need to deal with divergence which requires a proper regularization. To see
this, for Re$(\alpha^{\prime}s-2-z^{\prime})<0$, we use a change of variables

$x=\exp{\left(\frac{y}{\alpha^{\prime}s-2-z^{\prime}}\right)}$ (11)

to turn the amplitude (7) into

$\displaystyle\mathcal{A}(z^{\prime})$
$\displaystyle=\sum_{a=0}^{\infty}\binom{\alpha^{\prime}t-2}{a}(-1)^{a+1}\sum_{k=0}^{\infty}\frac{1}{k!}\int_{0}^{\infty}dy\
e^{y}\frac{((1+a)y)^{k}}{(\alpha^{\prime}s-2-z^{\prime})^{k+1}}$ (12)

where the binomial expansion was applied. Notice a slight difference from (8)
due to the minus signs. Now comes the divergence integral
$\int_{0}^{\infty}dy\ e^{y}y^{s}$ by which we can regularize it to be

$\int_{0}^{\infty}dy\ y^{s}e^{y}\coloneqq(-1)^{s+1}s!\qquad\text{for}\quad
s\in\mathbb{Z}^{+}\cup\\{0\\}.$ (13)

The symbol $\coloneqq$ signifies that the equality holds upon the
regularization. In this case, we assign the value for the integral via
analytic continuation of the parameter $n$ from

$\frac{1}{A^{n}}=\frac{1}{(n+1)!}\int_{0}^{\infty}dx\ x^{n-1}e^{-xA}.$ (14)

Accordingly, substituting (13) into (12), one would regain the same expression
of $\mathcal{A}(z^{\prime})$ as (9), hence providing the same on-shell
recursion relation (10).

##  3 On-shell Recursion Relations for $n-$point Open String Amplitudes

To generalize the investigation to a general $n$-point open string amplitude,
we consider a general expression

$\displaystyle\mathcal{A}(1,2,3,\ldots,n)=\int_{\mathcal{I}}\prod_{i=1}^{n}dz_{i}\frac{|z_{ab}z_{ac}z_{bc}|}{dz_{a}dz_{b}dz_{c}}\prod_{1\leq
i<j\leq n}|x_{i}-x_{j}|^{2\alpha^{\prime}k_{i}\cdot k_{j}}\mathcal{F}_{n}$
(15)

where $z_{ij}=x_{i}-x_{j}$ and $dz_{i}=dx_{i}$ for bosonic string theory and
$z_{ij}=x_{i}-x_{j}+\theta_{i}\theta_{j}$ and $dz_{i}=dx_{i}d\theta_{i}$ for
the supersymmetric case. The integral is subject to the integration region
$\mathcal{I}$ where $x_{1}<x_{2}\ldots<x_{n}$ which is associated to the group
factor tr($T_{1}T_{2}\ldots T_{n}$). The gauge symmetry $PSL(2,\mathbb{R})$
allow one to fix the position of three points denoted $z_{a},z_{b}$ and
$z_{c}$. A conventional choice is $x_{1}=0,x_{n-1}=1$ and $x_{n}=\infty$ for
the bosonic string as well as $\theta_{n-1}=\theta_{n}=0$ for for the
supersymmetric case.

The function $\mathcal{F}_{n}$ is a branch-free function that comes from the
operator product expansion of vertex operators depending on the external
states of the amplitude we consider. $\mathcal{F}_{n}=1$ for tachyons and
$\mathcal{F}_{n}=\text{exp}\Big{(}\sum_{i>j}\frac{\xi_{i}\cdot\xi_{j}}{(x_{i}-x_{j})^{2}}-\sum_{i\neq
j}\frac{\sqrt{\alpha^{\prime}}k_{i}\cdot\xi_{j}}{(x_{i}-x_{j})}\Big{)}\Big{|}_{\text{multilinear
in $\xi_{i}$}}$ for an $n$-gauge field amplitude with $n$ polarization vectors
$\xi_{i}$. In addition,
$\mathcal{F}_{n}=\int\prod_{i=1}^{n}d\eta_{i}\allowbreak\times\text{exp}\Big{[}\sum_{i\neq
j}\Big{(}\frac{\sqrt{\alpha^{\prime}}\eta_{i}(\theta_{i}-\theta_{j})(\xi_{i}\cdot
k_{j})-\eta_{i}\eta_{j}(\xi_{i}\cot\xi_{j})}{(x_{i}-x_{j}+\theta_{i}\theta_{j})}\Big{)}\Big{]}$
for the superstring amplitude where $\eta_{i}$ are Grassmann variables.

Alternatively, one could relate the Koba-Nielsen’s integral representation of
open string amplitudes (15) with multiple Gaussian hypergeometric functions,

$\displaystyle
B_{n}(\\{n\\})=\left(\prod_{i=1}^{n-3}\int_{0}^{1}dw_{i}\right)\prod_{j=1}^{n-3}w_{j}^{s_{12\ldots
j+1}+n_{j}}\prod_{l=j}^{n-3}\left(1-\prod_{k=j}^{l}w_{k}\right)^{s_{j+1,l+2}+n_{j+1,l+2}}$
(16)

where $s_{ij}=\alpha^{\prime}(k_{i}+k_{j})^{2}$ and $s_{12\ldots
i}=\alpha^{\prime}(k_{1}+k_{2}+\ldots+k_{i})^{2}$. The set $\\{n\\}$ contains
all integers $n_{i}$ and $n_{ij}$ appearing on the right-hand side of (16).
More precisely, the function (16) is known as generalized Kampé de Fériet
function [26]. To obtain the above expression, we applied a change of integral
variables

$x_{i}=\prod_{j=i-1}^{n-3}w_{j}\qquad\text{for}\quad i=2,3,\ldots,n-2$ (17)

to

$\int_{\mathcal{I}}\prod_{i=2}^{n-2}dx_{i}\ \prod_{1\leq i<j\leq
n}|x_{i}-x_{j}|^{2\alpha^{\prime}k_{i}\cdot k_{j}+\tilde{n}_{ij}}.$ (18)

Note that we have fixed $x_{1},x_{n-1}$ and $x_{n}$ to $0,1$ and $\infty$
respectively. The exponents $\tilde{n}_{ij}\in\mathbb{Z}$ which arise from the
external state-dependent term $\mathcal{F}_{n}$. They relate to the integers
$n_{i}$ and $n_{ij}$ via

$\displaystyle n_{ij}$
$\displaystyle=\tilde{n}_{ij}+\alpha^{\prime}(m_{i}^{2}+m_{j}^{2}),$
$\displaystyle i\leq j$ $\displaystyle n_{j}$
$\displaystyle=j-1+\sum_{l=1}^{j}\sum_{k>l}^{j+1}\tilde{n}_{lk}+\alpha^{\prime}\sum_{i=1}^{j+1}m_{i}^{2},$
$\displaystyle 1\leq j\leq n-3.$ (19)

More detail can be found in Chapter 7 of [27]. Consequently, the general
expression of color-ordered open string amplitudes reads

$\mathcal{A}(1,2,\ldots,n)=\sum_{I}\mathcal{K}_{I}B_{n}(\\{n^{I}\\})$ (20)

where the function $\mathcal{K}_{I}$ contains scalar products of momentum and
polarization vectors which are $k_{i}\cdot k_{j},k_{i}\cdot\xi_{j}$ and
$\xi_{i}\cdot\xi_{j}$ [28].

Rewriting the open string amplitude as (20) is useful to determine the pole
structure of the deformed amplitude $\mathcal{A}_{n}(z)$. This allows us to
formulate the on-shell recursion relations for $n$-point open string
amplitudes. When the momenta are shifted using (1), the function
$B_{n}(\\{n\\})$ becomes

$\displaystyle
B_{n}(\\{n\\},z)=\left(\prod_{i=1}^{n-3}\int_{0}^{1}dw_{i}\right)$
$\displaystyle\prod_{j=1}^{n-3}w_{j}^{s_{12\ldots j+1}+n_{j}-z_{j}}$
$\displaystyle\times\prod_{l=j}^{n-3}\left(1-\prod_{k=j}^{l}w_{k}\right)^{s_{j+1,l+2}+n_{j+1,l+2}}$
(21)

where $z_{i}=2\alpha^{\prime}z(q\cdot\sum_{j=2}^{i+1}k_{j})$. This implies
that there are $n-3$ poles generated from $z_{i}$ which refers to $n-3$
scattering channels. To determine the poles of $z_{i}$, we apply binomial
expansions to every term that contains $w_{i}$ in the product

$\prod_{j=1}^{n-3}\prod_{l=j}^{n-3}\left(1-\prod_{k=j}^{l}w_{k}\right)^{s_{j+1,l+2}+n_{j+1,l+2}}.$
(22)

For example, let’s first start by determining the pole regarding $z_{1}$.
Therefore, one needs to rewrite

$\displaystyle\prod_{l=1}^{n-3}\left(1-\prod_{k=1}^{l}w_{k}\right)^{s_{2,l+2}+n_{2,l+2}}=\prod_{l=1}^{n-3}\sum_{a_{l}=0}^{\infty}\binom{s_{2,l+2}+n_{2,l+2}}{a_{l}}(-1)^{a_{l}}w_{l}^{\sum_{i=l}^{n-3}a_{i}}$
(23)

using the binomial expansion. The expression involves $n-3$ parameters $a_{l}$
resulted from the expansions. This turns (21) to be

$\displaystyle B_{n}(\\{n\\},z)=$
$\displaystyle\prod_{l=1}^{n-3}\bigg{(}\sum_{a_{l}=0}^{\infty}\binom{s_{2,l+2}+n_{2,l+2}}{a_{l}}\bigg{)}(-1)^{\sum_{i=1}^{n-3}a_{i}}$
$\displaystyle\times\bigg{[}\int_{0}^{1}dw_{1}w_{1}^{s_{12}+n_{1}+\sum_{i=1}^{n-3}a_{i}-z_{1}}\bigg{]}\prod_{i=2}^{n-3}\int_{0}^{1}dw_{i}$
$\displaystyle\times\prod_{j=2}^{n-3}w_{j}^{s_{12\ldots
j+1}+n_{j}+\sum_{i=j}^{n-3}a_{i}-z_{j}}\prod_{l=2}^{n-3}\left(1-\prod_{k=2}^{l}w_{k}\right)^{s_{3,l+2}+n_{3,l+2}}.$
(24)

We then rename the sum over all $a_{l}$ to the new parameter, says

$\sum_{i=1}^{n-3}a_{i}=m$ (25)

which runs from zero to infinity. This index $m$ labels the poles from
$z_{1}$. Notice that the integral in the square bracket of (24) in the second
line produces the simple poles characterized by the index $m$

$\frac{(-1)}{z_{1}-(s_{12}+n_{1}+m+1)}.$ (26)

As a result, one can compute the residue

$\displaystyle\sum_{m}\text{Res}_{z=z^{*}_{m}}\bigg{(}\frac{B_{n}(\\{n\\},z)}{z}\bigg{)}=\sum_{m}\frac{\mathcal{B}_{1}(\\{n\\},m,z^{*}_{m})}{s_{12}+n_{1}+m+1}$
(27)

where $z^{*}_{m}=(s_{12}+n_{1}+m-1)/(2\alpha^{\prime}q\cdot k_{2})$. The
function $\mathcal{B}_{1}(\\{n\\},m,z)$ is defined as

$\displaystyle\prod_{l=1}^{n-4}\left(\sum_{a_{l}=0}^{\infty}\binom{s_{2,l+2}+n_{2,l+2}}{a_{l}}\right)\binom{s_{2,n-1}+n_{2,n-1}}{m-\sum_{l=1}^{n-4}a_{l}}(-1)^{m+1}\prod_{i=2}^{n-3}\int_{0}^{1}dw_{i}$
$\displaystyle\times\prod_{j=2}^{n-3}w_{j}^{s_{12\ldots
j+1}+n_{j}+m-\sum_{i=1}^{j-1}a_{i}-z_{j}}\prod_{l=2}^{n-3}\left(1-\prod_{k=2}^{l}w_{k}\right)^{s_{3,l+2}+n_{3,l+2}}.$
(28)

Note that $\mathcal{B}_{1}[n_{i},n_{ij},z]$ is evaluated at $z=z^{*}_{m}$ in
(27). The denominators of the equation (27) suggest the mass spectrum of
intermediate states with momentum $k_{1}+k_{2}$. What we need to do next is to
generalize this kind of calculation to involve all poles corresponding to all
$z_{j}$.

Similar to the previous calculation, to determine the poles associated with
$z_{i}$ for $i=1,2,\ldots,n-3$, one requires a binomial expansion to (22) to
expand all the terms that have $w_{i}$. These expansions would generate
$(n-2-i)i$ summing indices. As for the previous case, we had $n-3$ indices for
$i=1$.

For convenience, we will assign the summing index $a_{jl}$ for $j\leq l$ when
expanding the polynomial

$\displaystyle\left(1-\prod_{k=j}^{l}w_{k}\right)^{\tilde{s}_{j+1,l+2}}=\sum_{a_{jl}=0}^{\infty}\binom{\tilde{s}_{j+1,l+2}}{a_{jl}}(-1)^{a_{jl}}\left(\prod_{k=j}^{l}w_{k}\right)^{a_{jl}}$
(29)

where $\tilde{s}_{j+1,l+2}\equiv s_{j+1,l+2}+n_{j+1,l+2}$ for short. We will
use the above expansion to the product (22) for which $j\leq i\leq l$ (Note
that $i$ refers to the index where the poles $z_{i}$ will be determined). This
gives

$\displaystyle\prod_{j\leq l}$
$\displaystyle\left(1-\prod_{k=j}^{l}w_{k}\right)^{\tilde{s}_{j+1,l+2}}=$
$\displaystyle\prod_{\begin{subarray}{c}j\leq l\\\
i\notin[j,l]\end{subarray}}\left(1-\prod_{k=j}^{l}w_{k}\right)^{\tilde{s}_{j+1,l+2}}\prod_{\begin{subarray}{c}j\leq
l\\\
i\in[j,l]\end{subarray}}\left(\sum_{a_{jl}=0}^{\infty}\binom{\tilde{s}_{j+1,l+2}}{a_{jl}}(-1)^{a_{jl}}\left(\prod_{k=j}^{l}w_{k}\right)^{a_{jl}}\right).$
(30)

Note that we only apply the expansions for the case where $i\in[j,l]$ as
discussed. Accordingly, we can write (21) as

$\displaystyle B_{n}(\\{n\\},z)=$
$\displaystyle\left(\prod_{\begin{subarray}{c}k=1\\\ k\neq
i\end{subarray}}^{n-3}\int_{0}^{1}dw_{k}\right)\prod_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{n-3}w_{j}^{s_{12\ldots
j+1}+n_{j}-z_{j}}\prod_{\begin{subarray}{c}j\leq l\\\
i\notin[j,l]\end{subarray}}\left(1-\prod_{k=j}^{l}w_{k}\right)^{\tilde{s}_{j+1,l+2}}$
$\displaystyle\times\prod_{\begin{subarray}{c}j\leq l\\\
i\in[j,l]\end{subarray}}\left(\sum_{a_{jl}=0}^{\infty}\binom{\tilde{s}_{j+1,l+2}}{a_{jl}}(-1)^{a_{jl}}\left(\prod_{\begin{subarray}{c}k=j\\\
k\neq i\end{subarray}}^{l}w_{k}\right)^{a_{jl}}\right)$
$\displaystyle\times\int_{0}^{1}dw_{i}\ w_{i}^{s_{12\ldots
i+1}+n_{i}+\sum_{j\leq l}a_{jl}-z_{i}}$ (31)

Notice that we factor out the variable $w_{i}$ from the others in the last
line. Integrating out the $w_{i}$-variable turn (31) into

$\displaystyle
B_{n}(\\{n\\},z)=\sum_{m}\frac{\mathcal{B}_{i}(\\{n\\},m,z)}{z_{i}-(s_{12\ldots
i+1}+n_{i}+m+1)}$ (32)

where

$\displaystyle\mathcal{B}_{i}(\\{n\\},m,z)=$
$\displaystyle\left(\prod_{\begin{subarray}{c}k=1\\\ k\neq
i\end{subarray}}^{n-3}\int_{0}^{1}dw_{k}\right)\prod_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{n-3}w_{j}^{s_{12\ldots
j+1}+n_{j}-z_{j}}\prod_{\begin{subarray}{c}j\leq l\\\
i\notin[j,l]\end{subarray}}\left(1-\prod_{k=j}^{l}w_{k}\right)^{\tilde{s}_{j+1,l+2}}$
$\displaystyle\times(-1)^{m+1}\prod_{\begin{subarray}{c}j\leq l\\\
i\in[j,l]\\\
(j,l)\neq(1,n-3)\end{subarray}}\left(\sum_{a_{jl}=0}^{\infty}\binom{\tilde{s}_{j+1,l+2}}{a_{jl}}\left(\prod_{\begin{subarray}{c}k=j\\\
k\neq i\end{subarray}}^{l}w_{k}\right)^{a_{jl}}\right)$
$\displaystyle\times\binom{\tilde{s}_{2,n-1}}{m-\sum_{(j,l)\neq(1,n-3)}a_{jl}}\left(\prod_{k=1}^{n-3}w_{k}\right)^{m-\sum_{(j,l)\neq(1,n-3)}a_{jl}}.$
(33)

We denote the index $m$ as sum of all indices $a_{jl}$

$m=\sum_{\begin{subarray}{c}j\leq l\\\ i\in[j,l]\end{subarray}}a_{jl}.$ (34)

Accordingly, we need to write one index out of all the $a_{ij}$’s in terms of
$m$. Our choice is $a_{1,n-3}$ as this index always appears in the formulation
regardless of the index $i$ we consider. Remember that when $i=1$, the
expression for $\mathcal{B}_{1}(\\{n\\},m,z)$ matches with that presented in
(28).

Now, one can obtain the residue

$\displaystyle\sum_{z_{\text{pole}}}\text{Res}_{z=z_{\text{pole}}}\bigg{(}\frac{B_{n}(\\{n\\},z)}{z}\bigg{)}=\sum_{i=1}^{n-3}\sum_{m=0}^{\infty}\frac{\mathcal{B}_{i}(\\{n\\},m,z^{*}_{im})}{s_{12\ldots
i+1}+n_{i}+m+1}$ (35)

where

$z^{*}_{im}=\frac{s_{12\ldots
i+1}+n_{i}+m+1}{2\alpha^{\prime}q\cdot\sum_{j=2}^{i+1}k_{j}}.$ (36)

Consequently, the general expression for color-ordered open string amplitude
is then written as

$\displaystyle\mathcal{A}(1,2,3,\ldots,n)=\sum_{i=1}^{n-3}\sum_{m=0}^{\infty}\sum_{I}\mathcal{K}_{I}(z^{*}_{im})\frac{\mathcal{B}_{i}(\\{n^{I}\\},m,z^{*}_{im})}{s_{12\ldots
i+1}+n_{i}+m+1}.$ (37)

Note that the function $\mathcal{K}_{I}$ is also evaluated at $z^{*}_{im}$ as
well since the function is $z$-dependent when the momenta are shifted.

It is clear that the denominators in the expression (37) correspond to
propagators of the on-shell intermediate string states. For a specific example
of $n$-tachyon scattering, $n_{i}=-2$. The denominators suggest the full
spectrum of open strings, i.e.

$(k_{1}+k_{2}+\ldots+k_{i+1})^{2}+\frac{(m-1)}{\alpha^{\prime}}$ (38)

for a non-negative integer $m$. In the case of $n-1$ tachyons and one gauge
boson scattering assigning polarization for the vector boson as $\xi_{1}$, one
obtains $\tilde{n}_{1i}=-1$ for $i=2,3,\ldots,n$ through linearizing the
$\xi_{1}$ in $\mathcal{F}_{n}$. This also gives $n_{i}=-2$ providing the same
mass spectrum of propagating open strings. On the other hand, one can use the
requirement of the intermediate states to be on-shell to inversely identify
all $\tilde{n}_{ij}$ from the function $\mathcal{F}_{n}$ via (19).

The existence of scattering channel $s_{12\ldots i+1}$ implies that the
function (33) can be factorized into two objects referring to the amplitudes
of lower points. For the channel $s_{12\ldots i+1}$, one can write

$\displaystyle\mathcal{B}_{i}(\\{n\\},m,z)=$
$\displaystyle\prod_{\begin{subarray}{c}j\leq l\\\ i\in[j,l]\\\
(j,l)\neq(1,n-3)\end{subarray}}\left(\sum_{a_{jl}=0}^{\infty}\binom{\tilde{s}_{j+1,l+2}}{a_{jl}}\right)\binom{\tilde{s}_{2,n-1}}{m-\sum_{(j,l)\neq(1,n-3)}a_{jl}}$
$\displaystyle\times
B_{i,n}(\\{n\\},\\{a\\},m,z)\times\widetilde{B}_{i,n}(\\{n\\},\\{a\\},m,z)$
(39)

where

$\displaystyle B_{i,n}(\\{n\\},\\{a\\},m,z)=$
$\displaystyle\left(\prod_{j=1}^{i-1}\int_{0}^{1}dw_{j}\right)\prod_{k=1}^{i-1}w_{k}^{s_{12\ldots
k+1}+n_{k}-z_{k}+m-a_{ii}-\sum\limits_{\small{\begin{subarray}{c}j<l\\\
j,l=k+1\end{subarray}}}^{n-3}a_{jl}}$
$\displaystyle\times\prod_{l=k}^{i-1}\left(1-\prod_{r=k}^{l}w_{r}\right)^{s_{k+1,l+2}+n_{k+1,l+2}}$
(40)

and

$\displaystyle\widetilde{B}_{i,n}(\\{n\\},\\{a\\},m,z)=$
$\displaystyle\left(\prod_{j=i+1}^{n-3}\int_{0}^{1}dw_{j}\right)\prod_{k=i+1}^{n-3}w_{k}^{s_{12\ldots
k+1}+n_{k}-z_{k}+m-a_{ii}-\sum\limits_{\small{\begin{subarray}{c}j<l\\\
j,l=1\end{subarray}}}^{k-1}a_{jl}}$
$\displaystyle\times\prod_{l=k}^{n-3}\left(1-\prod_{r=k}^{l}w_{r}\right)^{s_{k+1,l+2}+n_{k+1,l+2}}.$
(41)

The functions $B_{i,n}(\\{n\\},\\{a\\},m,z)$ and
$\widetilde{B}_{i,n}(\\{n\\},\\{a\\},m,z)$ are in the forms of open string
amplitudes with $i+1$ and $n-i$ external legs respectively. This signifies the
fact that the on-shell recursion relations boil down a scattering amplitude in
terms of lower-point amplitudes.

##  4 Conclusions

In conclusion, we have developed a systematic approach to derive on-shell
recursion relations for open string amplitudes. The derivation, which is based
on the same technique initiated by Britto et al. [18, 19], starts from
applying the BCFW shift to an open string amplitude written in terms of
multiple Gaussian hypergeometric functions. The shift generates simple poles
corresponding to scattering channels of intermediate states. Using the residue
theorem, we expressed a general expression for $n-$point open string on-shell
recursion relations as in (37).

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