# The Mahler measure of the Rudin-Shapiro polynomials

Tamás Erdélyi  Department of Mathematics, Texas A&M University, College
Station, Texas 77843  terdelyimath.tamu.edu

Littlewood polynomials are polynomials with each of their coefficients in
$\\{-1,1\\}$. A sequence of Littlewood polynomials that satisfies a remarkable
flatness property on the unit circle of the complex plane is given by the
Rudin-Shapiro polynomials. It is shown in this paper that the Mahler measure
and the maximum modulus of the Rudin-Shapiro polynomials on the unit circle of
the complex plane have the same size. It is also shown that the Mahler measure
and the maximum norm of the Rudin-Shapiro polynomials have the same size even
on not too small subarcs of the unit circle of the complex plane. Not even
nontrivial lower bounds for the Mahler measure of the Rudin Shapiro
polynomials have been known before.

Rudin-Shapiro polynomials, Littlewood polynomials, Mahler measure

††support: 2010 Mathematics Subject Classifications. Primary: 11C08, 11B75
11P99; Secondary: 41A10, 05D99, 33E99.

## 1\. Introduction

Let $\alpha<\beta$ be real numbers. The Mahler measure
$M_{0}(Q,[\alpha,\beta])$ is defined for bounded measurable functions $Q$
defined on $[\alpha,\beta]$ as

$M_{0}(Q,[\alpha,\beta]):=\exp\left(\frac{1}{\beta-\alpha}\int_{\alpha}^{\beta}{\log|Q(e^{it})|\,dt}\right)\,.$

It is well known, see [17], for instance, that

$M_{0}(Q,[\alpha,\beta])=\lim_{q\rightarrow 0+}{M_{q}(Q,[\alpha,\beta])}\,,$

where

$M_{q}(Q,[\alpha,\beta]):=\left(\frac{1}{\beta-\alpha}\int_{\alpha}^{\beta}{\left|Q(e^{it})\right|^{q}\,dt}\right)^{1/q}\,,\qquad
q>0\,.$

It is a simple consequence of the Jensen formula that

$M_{0}(Q):=M_{0}(Q,[0,2\pi])=|c|\prod_{k=1}^{n}{\max\\{1,|z_{k}|\\}}$

for every polynomial of the form

$Q(z)=c\prod_{k=1}^{n}{(z-z_{k})}\,,\qquad c,z_{k}\in{\mathbb{C}}\,.$

See [3, p. 271] or [2, p. 3], for instance. Let
$D:=\\{z\in{\mathbb{C}}\\}:|z|<1\\}$ denote the open unit disk of the complex
plane. Let $\partial D:=\\{z\in{\mathbb{C}}\\}:|z|=1\\}$ denote the unit
circle of the complex plane.

Finding polynomials with suitably restricted coefficients and maximal Mahler
measure has interested many authors. The classes

${\Cal{L}}_{n}:=\left\\{p:p(z)=\sum_{k=0}^{n}{a_{k}z^{k}}\,,\quad
a_{k}\in\\{-1,1\\}\right\\}$

of Littlewood polynomials and the classes

${\Cal{K}}_{n}:=\left\\{p:p(z)=\sum_{k=0}^{n}{a_{k}z^{k}}\,,\quad
a_{k}\in{\mathbb{C}},\enskip|a_{k}|=1\right\\}$

of unimodular polynomials are two of the most important classes considered.
Observe that ${\Cal{L}}_{n}\subset{\Cal{K}}_{n}$ and

$M_{0}(Q)=M_{0}(Q,[0,2\pi])\leq M_{2}(Q,[0,2\pi])=\sqrt{n+1}$

for every $Q\in{\Cal{K}}_{n}$. Beller and Newman [1] constructed unimodular
polynomials $Q_{n}\in{\Cal{K}}_{n}$ whose Mahler measure $M_{0}(Q,[0,2\pi])$
is at least $\sqrt{n}-c/\log n$.

Section 4 of [2] is devoted to the study of Rudin-Shapiro polynomials.
Littlewood asked if there were polynomials $p_{n_{k}}\in{\Cal{L}}_{n_{k}}$
satisfying

$c_{1}\sqrt{n_{k}+1}\leq|p_{n_{k}}(z)|\leq c_{2}\sqrt{n_{k}+1}\,,\qquad
z\in\partial D\,,$

with some absolute constants $c_{1}>0$ and $c_{2}>0$, see [2, p. 27] for a
reference to this problem of Littlewood. To satisfy just the lower bound, by
itself, seems very hard, and no such sequence $(p_{n_{k}})$ of Littlewood
polynomials $p_{n_{k}}\in{\Cal{L}}_{n_{k}}$ is known. A sequence of Littlewood
polynomials that satisfies just the upper bound is given by the Rudin-Shapiro
polynomials. The Rudin-Shapiro polynomials appear in Harold Shapiro’s 1951
thesis [20] at MIT and are sometimes called just Shapiro polynomials. They
also arise independently in Golay’s paper [16]. They are remarkably simple to
construct and are a rich source of counterexamples to possible conjectures.

The Rudin-Shapiro polynomials are defined recursively as follows:

$P_{0}(z):=1\,,\qquad Q_{0}(z):=1\,,$

and

$\begin{split}P_{n+1}(z)&:=P_{n}(z)+z^{2^{n}}Q_{n}(z)\,,\cr
Q_{n+1}(z)&:=P_{n}(z)-z^{2^{n}}Q_{n}(z)\,,\cr\end{split}$

for $n=0,1,2,\ldots\,.$ Note that both $P_{n}$ and $Q_{n}$ are polynomials of
degree $N-1$ with $N:=2^{n}$ having each of their coefficients in
$\\{-1,1\\}$. It is well known and easy to check by using the parallelogram
law that

$|P_{n+1}(z)|^{2}+|Q_{n+1}(z)|^{2}=2(|P_{n}(z)|^{2}+|Q_{n}(z)|^{2})\,,\qquad
z\in\partial D\,.$

Hence

$|P_{n}(z)|^{2}+|Q_{n}(z)|^{2}=2^{n+1}=2N\,,\qquad z\in\partial D\,.$ $None$

It is also well known (see Section 4 of [2], for instance), that

$|Q_{n}(z)|=|P_{n}(-z)|\,,\qquad z\in\partial D\,.$ $None$

Peter Borwein’s book [2] presents a few more basic results on the Rudin-
Shapiro polynomials. Cyclotomic properties of the Rudin-Shapiro polynomials
are discussed in [8]. Obviously $M_{2}(P_{n},[0,2\pi])=2^{n/2}$ by the
Parseval formula. In 1968 Littlewood [18] evaluated $M_{4}(P_{n},[0,2\pi])$
and found that $M_{4}(P_{n},[0,2\pi])\sim(4^{n+1}/3)^{1/4}$. Rudin-Shapiro
like polynomials in $L_{4}$ on the unit circle of the complex plane are
studied in [6]. In 1980 Saffari conjectured that

$M_{q}(P_{n},[0,2\pi])\sim\frac{2^{(n+1)/2}}{(q/2+1)^{1/q}}$

for all even integers $q>0$, perhaps for all real $q>0$. This conjecture was
proved for all even values of $q\leq 52$ by Doche [11] and Doche and Habsieger
[12].

Despite the simplicity of their definition not much is known about the Rudin-
Shapiro polynomials. It is shown in this paper that the Mahler measure and the
maximum modulus of the Rudin-Shapiro polynomials on the unit circle of the
complex plane have the same size. A consequence of this result is also proved.
It is also shown in this paper that the Mahler measure and the maximum norm of
the Rudin-Shapiro polynomials have the same size even on not too small subarcs
of the unit circle of the complex plane. Not even nontrivial lower bounds for
the Mahler measure of the Rudin-Shapiro polynomials has been known before.

P. Borwein and Lockhart [7] investigated the asymptotic behavior of the mean
value of normalized $L_{p}$ norms of Littlewood polynomials for arbitrary
$p>0$. They proved that

$\lim_{n\rightarrow\infty}{\frac{1}{2^{n+1}}\,\sum_{f\in{\Cal{L}}_{n}}{\frac{(M_{q}(f,[0,2\pi]))^{q}}{n^{q/2}}}}=\Gamma\left(1+\frac{q}{2}\right)\,.$

An analogue of this result does not seem to be known for $q=0$ (the Mahler
measure). However, the recent paper [9] paper establishes beautiful results on
the average Mahler measure and $L_{q}({\partial D})$ norms of unimodular
polynomials (polynomials with complex coefficients of modulus $1$) of degree
$n$.

## 2 Main Theorems

Our first theorem states that the Mahler measure and the maximum norm of the
Rudin-Shapiro polynomials on the unit circle of the complex plane have the
same size.

###### Theorem 2.1

Let $P_{n}$ and $Q_{n}$ be the $n$-th Rudin-Shapiro polynomials defined in
Section 1. There is an absolute constant $c_{1}>0$ such that

$M_{0}(P_{n},[0,2\pi])=M_{0}(Q_{n},[0,2\pi])\geq c_{1}\sqrt{N}\,,$

where

$N:=2^{n}=\text{\rm deg}(P_{n})+1=\text{\rm deg}(Q_{n})+1\,.$

By following the line of our proof, it is easy to verify that $c_{1}=e^{-227}$
is an appropriate choice in Theorem 2.1. To formulate our next theorem we
define

$\widetilde{P}_{n}:=2^{-(n+1)/2}P_{n}\qquad\text{and}\qquad\widetilde{Q}_{n}:=2^{-(n+1)/2}Q_{n}\,.$
$None$

By using the above normalization, (1.1) can be rewritten as

$|\widetilde{P}_{n}(z)|^{2}+|\widetilde{Q}_{n}(z)|^{2}=1\,,\qquad z\in\partial
D\,.$ $None$

Let

$I_{q}(\widetilde{P}_{n}):=(M_{q}(\widetilde{P}_{n},[0,2\pi]))^{q}:=\frac{1}{2\pi}\int_{0}^{2\pi}{|\widetilde{P}_{n}(e^{i\tau})|^{q}\,d\tau}\,,\qquad
q>0\,.$

The following result on the moments of the Rudin-Shapiro polynomials is a
simple consequence of Theorem 2.1.

###### Theorem 2.2

There is a constant $L<\infty$ independent of $n$ such that

$\sum_{k=1}^{\infty}{\frac{I_{k}(\widetilde{P}_{n})}{k}}<L\,,\qquad
n=0,1,\ldots\,.$

Our final result states that the Mahler measure and the maximum norm of the
Rudin-Shapiro polynomials have the same size even on not too small subarcs of
the unit circle of the complex plane.

###### Theorem 2.3

There is an absolute constant $c_{2}>0$ such that

$M_{0}(P_{n},[\alpha,\beta])\geq c_{2}\sqrt{N}\,,\qquad N:=2^{n}=\text{\rm
deg}(P_{n})+1\,,$

for all $n\in{\mathbb{N}}$ and for all $\alpha,\beta\in{\mathbb{R}}$ such that

$\frac{32\pi}{N}\leq\frac{(\log N)^{3/2}}{N^{1/2}}\leq\beta-\alpha\leq
2\pi\,.$

The same upper bound holds for $M_{0}(Q_{n},[\alpha,\beta])$.

It looks plausible that Theorem 2.3 holds whenever
$32\pi/N\leq\beta-\alpha\leq 2\pi\,,$ but we do not seem to be able to handle
the case $32\pi/N\leq\beta-\alpha\leq(\log N)^{3/2}N^{-1/2}$ in this paper.

## 3\. Lemmas

A key to the proof of Theorem 2.1 is the following observation which is a
straightforward consequence of the definition of the Rudin-Shapiro polynomials
$P_{n}$ and $Q_{n}$.

###### Lemma 3.1

Let $n\geq 2$ be an integer, $N:=2^{n}$, and and let

$z_{j}:=e^{it_{j}}\,,\quad t_{j}:=\frac{2\pi j}{N}\,,\quad
j\in{\mathbb{Z}}\,.$

We have

$\begin{split}P_{n}(z_{j})&=2P_{n-2}(z_{j})\,,\qquad j=2u\,,\enskip
u\in{\mathbb{Z}}\,,\cr P_{n}(z_{j})&=(-1)^{(j-1)/2}2i\,Q_{n-2}(z_{j})\,,\qquad
j=2u+1\,,\enskip u\in{\mathbb{Z}}\,,\cr\end{split}$

where $i$ is the imaginary unit.

Another key to the proof of Theorem 2.1 is Theorem 1.3 from [15]. Let
${\Cal{P}}_{N}$ be the set of all polynomials of degree at most $N$ with real
coefficients.

###### Lemma 3.2

Assume that $n,m\geq 1$,

$0<\tau_{1}\leq\tau_{2}\leq\cdots\leq\tau_{m}\leq
2\pi\,,\quad\tau_{0}:=\tau_{m}-2\pi\,,\quad\tau_{m+1}:=\tau_{1}+2\pi\,.$

Let

$\delta:=\max\\{\tau_{1}-\tau_{0},\tau_{2}-\tau_{1},\ldots,\tau_{m}-\tau_{m-1}\\}\,.$

For every $A>0$ there is a $B>0$ depending only on $A$ such that

$\sum_{j=1}^{m}{\frac{\tau_{j+1}-\tau_{j-1}}{2}\,\log|P(e^{i\tau_{j}})|}\leq\int_{0}^{2\pi}{\log|P(e^{i\tau})|\,d\tau}+B$

for all $P\in{\Cal{P}}_{N}$ and $\delta\leq AN^{-1}$. Moreover, the choice
$B=9A^{2}$ is appropriate.

Our next lemma can be proved by a routine zero counting argument. Let
${\Cal{T}}_{k}$ be the set of all real trigonometric polynomials of degree at
most $k$.

###### Lemma 3.3

For $k\in{\mathbb{N}}$, $M>0$, and $\alpha\in{\mathbb{R}}$, let
$T\in{\Cal{T}}_{k}$ be defined by

$T(t)=\frac{M}{2}\,(1-\cos(k(t-t_{0})))=M\,\sin^{2}\left(\frac{k(t-t_{0})}{2}\right)\,.$
$None$

Let $a\in{\mathbb{R}}$ be fixed. Assume that $S\in{\Cal{T}}_{k}$ satisfies
$S(a)=T(a)>0$ and $0\leq S(t)\leq M$ holds for all $t\in{\mathbb{R}}$. Then

(i) $S(t)>T(t)$ holds for all $t\in(y,a)$ if $T$ is increasing on $(y,a)$.

(ii) $S(t)>T(t)$ holds for all $t\in(a,y)$ if $T$ is decreasing on $(a,y)$.

###### Demonstration Proof of Lemma 3.3

If the lemma were false then $S-T\in{\Cal{T}}_{k}$ would have at least $2k+1$
zeros in a period, by counting multiplicities. ∎

A straightforward consequence of Lemma 3.3 is the following. For the sake of
brevity let

$\gamma:=\sin^{2}(\pi/8)=\frac{1}{2}(1-\cos(\pi/4))\,.$

###### Lemma 3.4

Assume that $S\in{\Cal{T}}_{k}$ satisfies $0\leq S(t)\leq M$ for all
$t\in{\mathbb{R}}$. Let $a\in{\mathbb{R}}$ and assume that
$S(a)\geq(1-\gamma)M$. Then

$S(t)\geq\gamma M\,,\quad
t\in\left[a-\delta,a+\delta\right]\,,\quad\delta:=\frac{\pi}{2k}\,.$

###### Demonstration Proof of Lemma 3.4

Let $T\in{\Cal{T}}_{k}$ be defined by (3.1). Pick a $t_{0}\in{\mathbb{R}}$
such that $S(a)=T(a)$ and $T^{\prime}(a)\geq 0$. Now observe that $T$ is
increasing on $[a-\delta,a+\delta]$ and $T(a-\delta)\geq\gamma M$, and Lemma
3.3 (i) gives the lower bound of the lemma for all $t\in[a-\delta,a]$. Now
pick a $t_{0}\in{\mathbb{R}}$ such that $S(a)=T(a)$ and $T^{\prime}(a)\leq 0$.
Now observe that $T$ is decreasing on $[a,a+\delta]$ and
$T(a+\delta)\geq\gamma M$, and hence Lemma 3.3 (ii) gives the lower bound of
the lemma for all $t\in[a,a+\delta]$. ∎

Combining Lemmas 3.1 and 3.4 we easily obtain the following.

###### Lemma 3.5

Let $P_{n}$ and $Q_{n}$ be the $n$-th Rudin-Shapiro polynomials. Let
$N:=2^{n}$ and $\gamma:=\sin^{2}(\pi/8)$. Let

$z_{j}:=e^{it_{j}}\,,\quad t_{j}:=\frac{2\pi j}{N}\,,\qquad
j\in{\mathbb{Z}}\,.$

We have

$\max\\{|P_{n}(z_{j})|^{2},|P_{n}(z_{j+r})|^{2}\\}\geq\gamma 2^{n+1}=2\gamma
N\,,\quad r\in\\{-1,1\\}\,,$

for every $j=2u$, $u\in{\mathbb{Z}}$.

Lemma 3.5 tells us that the modulus of the Rudin-Shapiro polynomials $P_{n}$
is certainly larger than $\sqrt{2\gamma N}$ at least at one of any two
consecutive $N$-th root of unity, where $N:=2^{n}$. This is a crucial
observation of this paper, and despite its simplicity it does not seem to have
been observed before in the literature or elsewhere. Moreover, note that while
our Theorems 2.1 and 2.3 are proved with rather small multiplicative positive
absolute constants, Lemma 3.5 is stated with a quite decent explicit constant.

###### Demonstration Proof of Lemma 3.5

Let $k:=2^{n-2}$, $j=2u$, $u\in{\mathbb{Z}}$. We introduce the trigonometric
polynomial $S\in{\Cal{T}}_{k}$ by

$S(t):=|Q_{n-2}(e^{it})|^{2}\,,\qquad t\in{\mathbb{R}}\,.$ $None$

Recall that

$S(t)=|Q_{n-2}(e^{it})|^{2}\leq 2^{n-1}$ $None$

Assume that

$|P_{n}(z_{j})|^{2}<\gamma 2^{n+1}\,.$

Then (1.1) implies that

$|Q_{n}(z_{j})|^{2}>(1-\gamma)2^{n+1}\,.$ $None$

By Lemma 3.1 we have

$|P_{n-2}(z_{j})|^{2}=\frac{1}{4}\,|P_{n}(z_{j})|^{2}\,,$

and hence (1.1) implies that

$|Q_{n-2}(z_{j})|^{2}=\frac{1}{4}\,|Q_{n}(z_{j})|^{2}\,.$

Combining this with (3.3), we obtain

$|Q_{n-2}(z_{j})|^{2}=\frac{1}{4}\,|Q_{n}(z_{j})|^{2}>(1-\gamma)2^{n-1}\,.$

Hence, using Lemma 3.4 with $S\in{\Cal{T}}_{k}$ and $M:=2^{n-1}$ (recall that
(3.2) and (1.1) imply that $0\leq S(t)\leq 2^{n-1}$ for all
$t\in{\mathbb{R}}$), we can deduce that

$|Q_{n-2}(z_{j+r})|^{2}\geq\gamma 2^{n-1}\,,\qquad r\in\\{-1,1\\}\,.$

Finally we use Lemma 3.1 again to conclude that

$|P_{n}(z_{j+r})|^{2}=4|Q_{n-2}(z_{j+r})|^{2}\geq\gamma 2^{n+1}\,,\qquad
r\in\\{-1,1\\}\,.$

∎

To prove Theorem 2.3 we need Theorem 2.1 from [13]. We state it as our next
lemma by using a slightly modified notation.

###### Lemma 3.6

Let $\omega_{1}<\omega_{2}\leq\omega_{1}+2\pi\,,$

$\omega_{1}\leq\theta_{1}<\theta_{2}<\cdots<\theta_{\mu}\leq\omega_{2}\,,$
$\theta_{0}:=\omega_{1}-(\theta_{1}-\omega_{1})\,,\qquad\theta_{\mu+1}:=\omega_{2}+(\omega_{2}-\theta_{\mu})\,,$
$\delta:=\max\\{\theta_{1}-\theta_{0},\theta_{2}-\theta_{1},\ldots,\theta_{\mu+1}-\theta_{\mu}\\}\leq\frac{1}{2}\,\sin\frac{\omega_{2}-\omega_{1}}{2}\,.$

There is an absolute constant $c_{3}>0$ such that

$\sum_{j=0}^{\mu}{\frac{\theta_{j+1}-\theta_{j-1}}{2}\log|P(e^{i\theta_{j}})|}\leq\int_{\omega_{1}}^{\omega_{2}}{\log|P(e^{i\theta})|\,d\theta}+c_{3}E(N,\delta,\omega_{1},\omega_{2})$

for every polynomial $P$ of the form

$P(z)=\sum_{j=0}^{N}{b_{j}z^{j}}\,,\qquad b_{j}\in{\mathbb{C}}\,,\enskip
b_{0}b_{N}\neq 0\,,$

where

$E(N,\delta,\omega_{1},\omega_{2}):=(\omega_{2}-\omega_{1})N\delta+N\delta^{2}\log(1/\delta)+\sqrt{N\log
R}\left(\delta\log(1/\delta)+\frac{\delta^{2}}{\omega_{2}-\omega_{1}}\right)$

and $R:=|b_{0}b_{N}|^{-1/2}\|P\|_{\partial D}\,.$

Observe that $R$ appearing in the above theorem can be easily estimated by

$R\leq|b_{0}b_{N}|^{-1/2}(|b_{0}|+|b_{1}|+\cdots+|b_{N}|)\,.$

## 4\. Proof of Theorems 2.1, 2.2, and 2.3

###### Demonstration Proof of Theorem 2.1

Let, as before, $\gamma=\sin^{2}(\pi/8)$, $N:=2^{n}$, and

$z_{j}:=e^{it_{j}}\,,\quad t_{j}:=\frac{2\pi j}{N}\,,\quad
j\in{\mathbb{Z}}\,.$

By Lemma 3.5 we can choose

$\tau_{m}-2\pi=:\tau_{0}\leq 0<\tau_{1}<\tau_{2}<\cdots<\tau_{m}\leq 2\pi$

so that

$\\{\tau_{1},\tau_{2},\ldots,\tau_{m}\\}\subset\\{t_{1},t_{2},\ldots,t_{N}\\}\,,$
$None$ $\tau_{j+1}-\tau_{j}\leq\frac{4\pi}{N}\,,\qquad j=0,1,\ldots,m-1\,,$
$None$

and

$|P_{n}(e^{i\tau_{j}})|^{2}\geq\gamma 2^{n+1}\,,\qquad j=1,2,\ldots,m\,.$
$None$

Then the value

$\delta:=\max\\{\tau_{1}-\tau_{0},\tau_{2}-\tau_{1},\ldots,\tau_{m}-\tau_{m-1}$

appearing in Lemma 3.2 satisfies $\delta\leq AN^{-1}$ with $A=4\pi$. Let $B>0$
be chosen for $A:=4\pi$ according to Lemma 3.2. Combining
$P_{n}\in{\Cal{P}}_{N}$, (4.1), and Lemma 3.2, we conclude that

$\begin{split}2\pi\left(\frac{1}{2}\log
2^{n+1}+\frac{1}{2}\log\gamma\right)&\leq\sum_{j=1}^{m}{\frac{\tau_{j+1}-\tau_{j-1}}{2}\,\log|P_{n}(e^{i\tau_{j}})|}\cr&\leq\int_{0}^{2\pi}{\log|P_{n}(e^{i\tau})|\,d\tau}+B\,,\cr\end{split}$

and hence

$M_{0}(P_{n},[0,2\pi])\geq\exp(-B/(2\pi))\sqrt{2\gamma}\,2^{n/2}=c\sqrt{N}$

follows with the absolute constant $c:=\exp(-B/(2\pi))\sqrt{2\gamma}>0$.
Combining this with (1.2), we obtain

$M_{0}(Q_{n},[0,2\pi])\geq c2^{(n+1)/2}$

with the same absolute constant $c>0$. ∎

###### Demonstration Proof of Theorem 2.2

Recalling (2.2) and using the power series expansion of the function
$f(z):=\log(1-z)$ on $(-1,1)$, and the Monotone Convergence Theorem, we deduce
that

$\begin{split}\int_{0}^{2\pi}{\log|\widetilde{P}_{n}(e^{i\tau})|^{2}\,d\tau}&=\int_{0}^{2\pi}{\log(1-|\widetilde{Q}_{n}(e^{i\tau})|^{2})\,d\tau}=\int_{0}^{2\pi}{-\sum_{k=1}^{\infty}{\frac{|\widetilde{Q}_{n}(e^{i\tau})|^{2k}}{k}\,d\tau}}\cr&=\int_{0}^{2\pi}{-\sum_{k=1}^{\infty}{\frac{|\widetilde{P}_{n}(e^{i\tau})|^{2k}}{k}\,d\tau}}=-\sum_{k=1}^{\infty}{\int_{0}^{2\pi}{\frac{|\widetilde{P}_{n}(e^{i\tau})|^{2k}}{k}\,d\tau}}\cr&=-2\pi\sum_{k=1}^{\infty}{\frac{I_{2k}(\widetilde{P}_{n})}{k}}\,.\cr\end{split}$

Combining this with Theorem 2.1 gives that there is an $L<\infty$ independent
of $n$ such that

$\sum_{k=1}^{\infty}{\frac{I_{2k}(\widetilde{P}_{n})}{k}}<L\,.$

As $I_{k}(\widetilde{P}_{n})$ is a decreasing function of $k\in{\mathbb{N}}$,
the theorem follows. ∎

###### Demonstration Proof of Theorem 2.3

The theorem follows from Lemmas 3.5 and 3.6 in a straightforward fashion. Note
that

$(M_{0}(f,[\alpha,\beta]))^{\beta-\alpha}=(M_{0}(f,[\alpha,\gamma]))^{\gamma-\alpha}(M_{0}(f,[\gamma,\beta]))^{\beta-\gamma}\,,$

for all $\alpha<\gamma<\beta\leq\alpha+2\pi$ and for all functions $f$
continuous on $[\alpha,\beta]$. Hence, to prove the theorem, without loss of
generality we may assume that $\beta-\alpha\leq\pi$. Let, as before,
$\gamma=\sin^{2}(\pi/8)$, $N:=2^{n}$, and

$z_{j}:=e^{it_{j}}\,,\quad t_{j}:=\frac{2\pi j}{N}\,,\quad
j\in{\mathbb{Z}}\,.$

By Lemma 3.5 we can choose

$\tau_{m}-2\pi=:\tau_{0}\leq 0<\tau_{1}<\tau_{2}<\cdots<\tau_{m}\leq 2\pi$

so that (4.1), (4.2), and (4.3) hold. Let

$\\{\theta_{1}<\theta_{2}<\cdots<\theta_{\mu}\\}:=\\{\tau_{j}\in[\alpha,\beta]:j=1,2,\ldots,m\\}\,.$
$None$

The assumption on $N$ guarantees that the value of $\delta$ defined in Lemma
3.6 is at most $4\pi/N$ and

$\frac{4\pi}{N}\leq\frac{\beta-\alpha}{8}\leq\frac{1}{2}\sin\frac{\beta-\alpha}{2}\,.$

Observe also that $P_{n}\in{\Cal{L}}_{N-1}$, and hence when we apply Lemma 3.6
to $P_{n}$ we have $R\leq N$. By (4.1) we have

$|P_{n}(e^{i\theta_{j}})|^{2}\geq\gamma 2^{n+1}\,,\qquad j=0,1,\ldots,\mu\,.$

Applying Lemma 3.6 with $P:=P_{n}$, $N:=2^{n}$, and
$\\{\theta_{1}<\theta_{2}<\cdots<\theta_{\mu}\\}$ defined by (4.4) we obtain

$\begin{split}(\beta-\alpha)\left(\frac{1}{2}\log
2^{n+1}+\frac{1}{2}\log\gamma\right)&\leq\sum_{j=0}^{\mu}{\frac{\theta_{j+1}-\theta_{j-1}}{2}\log|P_{n}(e^{i\theta_{j}})|}\cr&\leq\int_{\alpha}^{\beta}{\log|P_{n}(e^{i\theta})|\,d\theta}+c_{3}E(N,4\pi/N,\alpha,\beta)\,,\cr\end{split}$

where the assumption

$\frac{(\log N)^{3/2}}{N^{1/2}}\leq\beta-\alpha\leq 2\pi$

implies that

$\begin{split}E(N,4\pi/N,\alpha,\beta)\leq&\,c_{4}\left(\frac{(\beta-\alpha)N}{N}+\frac{\log
N}{N}+\sqrt{N\log N}\left(\frac{\log
N}{N}+\frac{1}{N^{2}(\beta-\alpha)}\right)\right)\cr\leq&\,c_{5}(\beta-\alpha)\cr\end{split}$

with absolute constants $c_{4}>0$ and $c_{5}>0$. Hence

$M_{0}(P_{n},[\alpha,\beta])\geq\exp(-c_{3}c_{5})\,\sqrt{2\gamma}\,2^{n/2}=c\sqrt{N}$

with the absolute constant $c:=\exp(-c_{3}c_{5})\,\sqrt{2\gamma}>0$. Combining
this with (1.2), we obtain

$M_{0}(Q_{n},[\alpha,\beta])\geq c\sqrt{N}$

with the same absolute constant $c>0$. ∎

## 5\. Some Recent results on Fekete Polynomials

This section could be viewed as part of the introduction. As the results on
Fekete polynomials mentioned in this section are somewhat analogous to our new
results on the Rudin-Shapiro polynomials, we close this paper with this short
section.

For a prime number $p$ the $p$-th Fekete polynomial is defined as

$f_{p}(z):=\sum_{k=1}^{p-1}{\left(\frac{k}{p}\right)z^{k}}\,,$

where

$\left(\frac{k}{p}\right)=\begin{cases}1,\quad\text{if \enskip}x^{2}\equiv
k\enskip(\text{mod\,}p)\enskip\text{has a nonzero solution,}\\\
0,\quad\text{if \enskip}p\enskip\text{divides}\enskip k\,,\\\
-1,\quad\text{otherwise}\end{cases}$

is the usual Legendre symbol. Since $f_{p}$ has constant coefficient $0$, it
is not a Littlewood polynomial, but $g_{p}$ defined by $g_{p}(z):=f_{p}(z)/z$
is a Littlewood polynomial of degree $p-2$, and has the same Mahler measure as
$f_{p}$. Fekete polynomials are examined in detail in [2], [4], [5], [10]. In
[19] Montgomery proved that there are absolute constants $c_{1}>0$ and
$c_{2}>0$ such that

$c_{1}\sqrt{p}\log\log p\leq\max_{z\in\partial D}{|f_{p}(z)|}\leq
c_{2}\sqrt{p}\log p\,.$

In [15] we gave a lower bound for the Mahler measure of the Fekete
polynomials. Namely we showed that for every $\varepsilon>0$ there is a
constant $c_{\varepsilon}$ such that

$M_{0}(f_{p},[0,2\pi])\geq\left(\frac{1}{2}-\varepsilon\right)\sqrt{p}$

for all primes $p\geq c_{\varepsilon}$. One of the key lemmas in the proof of
this theorem is a remarkable property of the Fekete polynomials observed by
Gauss. It states that

$|f_{p}(z_{p}^{j})|=p^{1/2}\,,\qquad j=1,2,\ldots,p-1\,,$

and $f_{p}(1)=0$, where $z_{p}:=\exp(2\pi i/p)$ is the first $p$-th root of
unity. A simple proof of it is given in [2, pp. 37-38]. In [13] Theorem 1.2 is
extended to subarcs of the unit circle. It is shown in [E-11] that there is an
absolute constant $c_{1}>0$ such that

$M_{0}(f_{p},[\alpha,\beta])\geq c_{1}p^{1/2}$

for all prime numbers $p$ and for all $\alpha,\beta\in{\mathbb{R}}$ such that
$(\log p)^{3/2}p^{-1/2}\leq\beta-\alpha\leq 2\pi$. In [14] the author gave an
upper bound for the average value of $|f_{p}(z)|^{q}$ over any subarc $I$ of
the unit circle, valid for all sufficiently large primes $p$ and arbitrary
real exponents $q>0$. Namely, there is a constant $c_{2}(q,\varepsilon)$
depending only on $q>0$ and $\varepsilon>0$ such that

$M_{q}(f_{p},[\alpha,\beta])\leq c_{2}(q,\varepsilon)p^{1/2}\,,$

for all prime numbers $p$ and for all $\alpha,\beta\in{\mathbb{R}}$ such that
$\beta-\alpha\geq 2p^{-1/2+\varepsilon}$.

### 6\. Acknowledgment

The author thanks Peter Borwein, Stephen Choi, Michael Mossinghoff, and Bahman
Saffari for their careful reading of the paper and their comments on making
the paper better.

## References

  * 1 E. Beller and D.J. Newman,, An extremal problem for the geometric mean of polynomials, Proc. Amer. Math. Soc. 39 (1973), 313–317.
  * 2 P. Borwein, Computational Excursions in Analysis and Number Theory, Springer, 2002.
  * 3 P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer, 1995.
  * 4 P. Borwein and K.-K. S. Choi, Explicit merit factor formulae for Fekete and Turyn polynomials, Trans. Amer. Math. Soc. 354 (2002), no. 2 219–234.
  * 5 P. Borwein and K.-K. S. Choi, The average norm of polynomials of fixed height, Trans. Amer. Math. Soc. 359 (2007), no. 2 923–936.
  * 6 P. Borwein and M.J. Mossinghoff, Rudin-Shapiro like polynomials in $L_{4}$, Math. Comp. 69 (2000), 1157–1166.
  * 7 P. Borwein and R. Lockhart, The expected $L_{p}$ norm of random polynomials, Proc. Amer. Math. Soc. (2001), 1463–1472.
  * 8 J. Brillhart, J.S. Lemont, and P. Morton, Cyclotomic properties of the Rudin-Shapiro polynomials, J. Reine Angew. Math. (Crelle’s J.) 288 (1976), 37–65.
  * 9 K.-K. S. Choi and M.J. Mossinghoff, Average Mahler’s measure and Lp norms of unimodular polynomials, Pacific J. Math. 252 (2011), no. 1, 31–50.
  * 10 B. Conrey, A. Granville, B. Poonen, and K. Soundararajan, Zeros of Fekete polynomials, Ann. Inst. Fourier (Grenoble) 50 (2000), 865–884.
  * 11 Ch. Doche, Even moments of generalized Rudin-Shapiro polynomials, Math. Comp. 74 (2005), no. 252, 1923–1935.
  * 12 Ch. Doche and L. Habsieger, Moments of the Rudin-Shapiro polynomials, J. Fourier Anal. Appl. 10 (2004), no. 5, 497–505.
  * 13 T. Erdélyi, Sieve-type lower bounds for the Mahler measure of polynomials on subarcs, Computational Methods and Function Theory 11 (2011), 213–228.
  * 14 T. Erdélyi, Upper bounds for the Lq norm of Fekete polynomials on subarcs, Acta Arith. 153 (2012), no. 1, 81–91.
  * 15 T. Erdélyi and D. Lubinsky, Large sieve inequalities via subharmonic methods and the Mahler measure of Fekete polynomials, Canad. J. Math. 59 (2007), 730–741.
  * 16 M.J. Golay, Static multislit spectrometry and its application to the panoramic display of infrared spectra,, J. Opt. Soc. America 41 (1951), 468–472.
  * 17 G.H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, 1952.
  * 18 J.E. Littlewood, Some Problems in Real and Complex Analysis, Heath Mathematical Monographs, 1968.
  * 19 H.L. Montgomery, An exponential polynomial formed with the Legendre symbol, Acta Arith. 37 (1980), 375–380.
  * 20 H.S. Shapiro, Master thesis, MIT, 1951.


# On the matrix $q$-Kummer equation and its solutions

Ravi Dwivedi111E-mail<EMAIL_ADDRESS>
Govt. Naveen College Bhairamgarh
Bijapur (CG) 494450, India

###### Abstract

In the present paper, a general theory for the second-order matrix difference
equation of bilateral type is discussed. We introduced the matrix $q$-Kummer
equation of bilateral type and presented the $q$-Kummer matrix function as a
series solution. Later, we obtain the integral solutions of this Kummer
equation and solution at $\infty$. We also give a brief idea about the matrix
Gauss difference equation of bilateral type and its solutions.

Keywords: Basic hypergeometric function, Difference equations, Matrix
functional calculus.

AMS Subject Classification: 15A15, 39A13, 33D15.

##  1 Introduction

Most of the special functions are presented as a solution of either
differential equation or integral equation and hence they play a vital role in
the study of special functions. Due to a wide range of applications the Gauss
hypergeometric equation and Kummer hypergeometric equation obtained special
treatment from mathematician and physicist. These equations are studied from
various point of view, for example the $q$-analogue of these two equations are
given in [7] and [8] respectively; The matrix version of Gauss hypergeometric
function can be seen in [10, 11] and its $q$-analogue in [15].

In this paper, alongside the theory of the bilateral type matrix differential
equation, we have tried to develop the same for the bilateral type matrix
difference equation. In particular, we give a general form of a second order
matrix difference equation of bilateral type and analyse its solutions.
Furthermore, we define the ordinary and singular points of the $q$-Kummer
matrix equation of bilateral type and then present solutions on its singular
points. The treatment of the paper is as follows:

In section 2, we list some basic definitions and preliminaries that are needed
in the subsequent sections. In section 3, a general second order bilateral
type matrix $q$-difference equation is discussed. In section 4, the $q$-Kummer
matrix function is defined and its regions of convergence is deduced. We also
give matrix $q$-Kummer equation and various form of solutions is discussed. At
last, in section 5, we added a conclusion part which includes the scope of
this initial work.

##  2 Basic definitions and Preliminaries

The generalized hypergeometric function is defined by [6, 13, 16]

$\displaystyle{}_{r}F_{s}(z)={}_{r}F_{s}(\lambda_{1},\dots,\lambda_{r};\mu_{1},\dots,\mu_{s};z)=\sum_{n\geq
0}\frac{(\lambda_{1})_{n}\cdots(\lambda_{r})_{n}}{(\mu_{1})_{n}\cdots(\mu_{s})_{n}}\,\frac{z^{n}}{n!},$
(2.1)

where $\lambda_{1},\dots,\lambda_{r},\mu_{1},\dots,\mu_{s},z\in\mathbb{C}$
such that $\mu_{j}\neq 0,-1,-2,\dots,1\leq j\leq s$ and the Pochhammer symbol
$(\lambda)_{n}$ is defined as

$(\lambda)_{n}=\begin{cases}1,&\text{ if }n=0,\\\
\lambda(\lambda+1)\cdots(\lambda+n-1),&\text{ if }n\geq 1.\end{cases}$ (2.2)

The infinite series (2.1) converges absolutely in the following regions

  1. 1.

For $|z|<\infty$ if $r\leq s$;

  2. 2.

For $|z|<1$ if $r=s+1$;

  3. 3.

For $|z|=1$ if $r=s+1$ and
$\left(\sum_{j=1}^{s}\mu_{j}-\sum_{j=1}^{r}\lambda_{j}\right)>0$.

The differential equation satisfied by generalized hypergeometric function
(2.1) is

$\displaystyle\left[\theta(\theta+\mu_{1}-1)\cdots(\theta+\mu_{s}-1)-z(\theta+\lambda_{1})\cdots(\theta+\lambda_{r})\right]\,{}_{r}F_{s}(z)=0,$
(2.3)

where $\theta=z\frac{d}{dz}$.

A function $g_{q}$ with parameter $q$ is called the $q$-analogue of an
ordinary function $g$ if $g_{q}$ approaches to $g$ whenever $q\rightarrow 1$.
With this basic idea, one of the $q$-analogue of differential equation (2.3)
has been given by Jia _et. al._ , [8], as

$\displaystyle[\delta_{q}$
$\displaystyle(q^{\mu_{1}-1}\delta_{q}+[\mu_{1}-1]_{q})\cdots(q^{\mu_{s}-1}\delta_{q}+[\mu_{s}-1]_{q})$
$\displaystyle-z\,(q^{\lambda_{1}}\delta_{q}+[\lambda_{1}]_{q})\cdots(q^{\lambda_{r}}\delta_{q}+[\lambda_{r}]_{q})]u(z)=0,$
(2.4)

where $\delta_{q}u(z)=zD_{q}u(z)=z\,\frac{u(z)-u(qz)}{z-qz}$,
$[\lambda]_{q}=\frac{1-q^{\lambda}}{1-q}$. As $\lim_{q\rightarrow
1}\delta_{q}\rightarrow\theta$ and $\lim_{q\rightarrow
1}[\lambda]_{q}\rightarrow\lambda$, so the $q$-difference equation (2.4) will
tend to the ordinary differential equation (2.3) when $q\rightarrow 1$. One
can verify that the equation (2.4) has a solution

$\displaystyle u(z)$
$\displaystyle={}_{r}\phi_{s}(q^{\lambda_{1}},\dots,q^{\lambda_{r}};q^{\mu_{1}},\dots,q^{\mu_{s}};q;(1-q)^{1+s-r}z)$
$\displaystyle=\sum_{n\geq
0}\frac{(q^{\lambda_{1}};q)_{n}\cdots(q^{\lambda_{r}};q)_{n}}{(q^{\mu_{1}};q)_{n}\cdots(q^{\mu_{s}};q)_{n}}\,\frac{(1-q)^{n\,(1+s-r)}z^{n}}{(q;q)_{n}},$
(2.5)

where

$(q^{\lambda_{1}};q)_{n}=\begin{cases}1,&\text{if $n=0$,}\\\
(1-q^{\lambda_{1}})(1-q^{\lambda_{1}+1})\cdots(1-q^{\lambda_{1}+n-1}),&\text{if
$n\geq 1$}.\end{cases}$

The series in (2.5) converges absolutely for $\left|(1-q)^{1+s-r}z\right|<1$.

For $r=2$, $s=1$, $\lambda_{1}=\lambda$, $\lambda_{2}=\nu$ and $\mu_{1}=\mu$,
the equation (2.4) produce the $q$-analogue of Gauss hypergeometric equation
as

$\displaystyle[\delta_{q}\,(q^{\mu-1}\delta_{q}+[\mu-1]_{q})-z\,(q^{\lambda}\delta_{q}+[\lambda]_{q})\,(q^{\nu}\delta_{q}+[\nu]_{q})]\,u(z)=0.$
(2.6)

Further, for $r=1$, $s=1$, $\lambda_{1}=\lambda$ and $\mu_{1}=\mu$, the
equation (2.4) leads to the $q$-analogue of Kummer equation

$\displaystyle[\delta_{q}\,(q^{\mu-1}\delta_{q}+[\mu-1]_{q})-z\,(q^{\lambda}\delta_{q}+[\lambda]_{q})]\,u(z)=0,$
(2.7)

which can also be written as

$\displaystyle
q^{\nu}\,z\,D_{q}^{2}\,u(z)+\left([\nu]_{q}-q^{\lambda}\,z\right)D_{q}\,u(z)-[\lambda]_{q}u(z)=0.$
(2.8)

The detailed study on this particular $q$-Kummer equation has been done in
[8].

For a complex valued function $f(z$), the $q$-derivative is defined by [12]

$\displaystyle D_{q}f(z)=\frac{f(z)-f(qz)}{(1-q)z}$ (2.9)

and the $q$-derivative of product of two complex valued functions $f(z)$ and
$g(z)$ is given as

$\displaystyle D_{q}(f(z).g(z))$
$\displaystyle=f(qz)\,D_{q}\,g(z)+g(z)\,D_{q}\,f(z).$ (2.10)

By symmetry, we can interchange $f$ and $g$

$\displaystyle D_{q}(f(z).g(z))=f(z)\,D_{q}\,g(z)+g(qz)\,D_{q}\,f(z).$ (2.11)

The $q$-derivative of quotient of $f(z)$ and $g(z)$ is given by

$\displaystyle
D_{q}\left(\frac{f(z)}{g(z)}\right)=\frac{g(z)\,D_{q}\,f(z)-f(z)\,D_{q}\,g(z)}{g(z)\,g(qz)}.$
(2.12)

If we use (2.11) instead of (2.10), we get

$\displaystyle
D_{q}\left(\frac{f(z)}{g(z)}\right)=\frac{g(qz)\,D_{q}\,f(z)-f(qz)\,D_{q}\,g(z)}{g(z)\,g(qz)}.$
(2.13)

The definite $q$-integral of a function $g(u)$, for complex number $u$ and
$z$, is defined by [7, 12]

$\displaystyle\int_{0}^{z}g(u)\,d_{q}u=(1-q)\,\sum_{j\geq
0}z\,q^{j}\,g(zq^{j}).$ (2.14)

In general,

$\displaystyle\int_{0}^{z}g(u)\,d_{q}\,h(u)=\int_{0}^{z}g(u)\,D_{q}h(u)\,d_{q}u=\sum_{j\geq
0}g(zq^{j})\left(h(zq^{j})-h(zq^{j+1})\right).$ (2.15)

The $q$-analogue of by part integration is given by, [7, 12]

$\displaystyle\int_{0}^{z}g(qu)\,d_{q}\,h(u)=g(z)\,h(z)-g(0)\,h(0)-\int_{0}^{z}h(u)\,D_{q}g(u).$
(2.16)

Further, the $q$-integral of $g(u)$ on $[0,\infty)$ is given by

$\displaystyle\int_{0}^{\infty}g(u)\,d_{q}u=(1-q)\,\sum_{j=-\infty}^{\infty}q^{j}\,g(q^{j}).$
(2.17)

The $q$-derivative of a definite $q$-integral is given by the following
formula, [8]

$\displaystyle
D_{q}\left(\int_{0}^{\frac{1}{az^{k}}}g(z,u)\,d_{q}u\right)=\left(\int_{0}^{\frac{1}{az^{k}}}D_{q}g(z,u)\,d_{q}u\right)-\sum_{j=0}^{k-1}\frac{q^{j-k}}{a\,z^{k+1}}g\left(qz,\frac{q^{j-k}}{az^{k}}\right),$
(2.18)

where $a$, $z$ are complex numbers and $D_{q}$ is the $q$-derivative with
respect to $z$. Letting $q\rightarrow 1$, the formula (2.18) yields

$\displaystyle\frac{d}{dz}\left(\int_{0}^{\frac{1}{az^{k}}}g(z,u)\,d_{q}u\right)=\int_{0}^{\frac{1}{az^{k}}}\frac{\partial}{\partial
z}g(z,u)\,d_{q}u-\frac{k}{a\,z^{k}}\,g(z,\frac{1}{a\,z^{k}}).$ (2.19)

Let $P$ be a $p$-square complex matrix. Then, $\alpha(P)=\max\\{\,\Re(z)\mid
z\in\sigma(P)\,\\}$ and $\beta(P)=\min\\{\,\Re(z)\mid z\in\sigma(P)\,\\}$,
where $\Re(z)$ is the real part of $z\in\mathbb{C}$ and $\sigma(P)$ is the set
of all eigenvalues of $P$. The matrix $P\in\mathbb{C}^{p\times p}$ is called
positive stable if $\beta(P)>0$.

The $q$-analogue of a matrix $P$ is defined by [14]

$\displaystyle[P]_{q}=\frac{I-q^{P}}{1-q},\quad q^{P}=\exp(P\,\ln q),$ (2.20)

where $\ln$ is the principal branch of logarithmic function. The matrix
analogue of $q$-shifted factorial function, for $P\in\mathbb{C}^{p\times p}$,
is given by [14, 15]

$(q^{P};q)_{n}=\begin{cases}I,&\text{if $n=0$,}\\\
(I-q^{P})(I-q^{P+I})\cdots(I-q^{P+(n-1)I}),&\text{if $n\geq 1$}.\end{cases}$

Furthermore, we have

$(q^{P};q)_{\infty}=\lim_{n\rightarrow\infty}\prod_{k=0}^{n-1}(I-q^{P+kI})=\prod_{k=0}^{\infty}(I-q^{P+kI}),$

which is convergent for $|q|<1$, [14]. If $\|q^{P}\|<1$ and
$q^{-k}\notin\sigma(q^{P})$, $k\in\mathbb{Z}^{+}$ (set of positive integers),
then $(q^{P};q)_{\infty}$ converges invertibly, [14, 15, 17].

The $q$-gamma matrix function, for $P\in\mathbb{C}^{p\times p}$ such that
$\beta(P)>0$, is defined as [14]

$\Gamma_{q}(P)=\int_{0}^{\frac{1}{1-q}}u^{P-I}E_{q}(-qu)\ d_{q}u,$ (2.21)

where $E_{q}(z)$ is the $q$-exponential function defined by

$\displaystyle E_{q}(z)=\sum_{j=0}^{\infty}\frac{q^{j\choose
2}\,\left(z\right)^{j}}{[j]_{q}!}=(-(1-q)z;q)_{\infty}.$ (2.22)

Furthermore, if $P$ and $Q$ commute, then the $q$-beta matrix function is
given by [14]

$\displaystyle\mathcal{B}_{q}(P,Q)=\mathcal{B}_{q}(Q,P)$
$\displaystyle=\int_{0}^{1}(uq;q)_{\infty}\ (uq^{Q};q)_{\infty}^{-1}\
u^{P-I}d_{q}u$ $\displaystyle=\Gamma_{q}(P)\ \Gamma_{q}(Q)\
\Gamma_{q}^{-1}(P+Q),$ (2.23)

where inverse of $q$-gamma matrix function $\Gamma_{q}^{-1}(P)$ is defined by

$\Gamma_{q}^{-1}(P)=[P]_{q}\,[P+I]_{q}\,\cdots\,[P+(n-1)I]_{q}\,\Gamma_{q}^{-1}(P+nI),\quad
n=1,2,\dots.$ (2.24)

In view of the equation (2.24), one can write

$(q^{P};q)_{n}=(1-q)^{n}\,\Gamma_{q}^{-1}(P)\,\Gamma_{q}(P+nI),\quad
n=1,2,\dots.$ (2.25)

For detailed study on basic matrix functions, one can check these papers and
references therein [2, 5, 14, 15].

##  3 Matrix $q$-difference equations

Consider the following general matrix $q$-difference equation

$\displaystyle[\delta_{q}$
$\displaystyle(q^{Q_{1}-1}\delta_{q}+[Q_{1}-1]_{q})\cdots(q^{Q_{s}-1}\delta_{q}+[Q_{s}-1]_{q})$
$\displaystyle-z\,(q^{P_{1}}\delta_{q}+[P_{1}]_{q})\cdots(q^{P_{r}}\delta_{q}+[P_{r}]_{q})]W(z)=0,$
(3.1)

where $P_{1},\dots,P_{r},Q_{1},\dots,Q_{s}$ are matrices in
$\mathbb{C}^{p\times p}$ such that $q^{-k}\notin\sigma(q^{Q_{j}}),1\leq j\leq
s$, $k\in\mathbb{Z}^{+}$. One can check that the generalized basic
hypergeometric matrix function, defined by

$\displaystyle{}_{r}\phi_{s}(q^{P_{1}},\dots,q^{P_{r}};q^{Q_{1}},\dots,q^{Q_{s}};q;(1-q)^{1+s-r}z)$
$\displaystyle=\sum_{n\geq
0}(q^{P_{1}};q)_{n}\cdots(q^{P_{r}};q)_{n}(q^{Q_{1}};q)^{-1}_{n}\cdots(q^{Q_{s}};q)^{-1}_{n}\,\frac{(1-q)^{n\,(1+s-r)}z^{n}}{(q;q)_{n}},$
(3.2)

is the solution of the matrix $q$-difference equation (3.1). While proving
this, we have to directly or indirectly assume that the matrices
$P_{1},\dots,P_{r},Q_{1},\dots,Q_{s}$ are commutative. To reduce this
comprehensive assumption of commutativity, we introduce the concept of matrix
$q$-difference equations of bilateral type.

The bilateral type matrix differential equation has been introduced by Jódar
and Cortés [10, 11] and studied the Gauss hypergeometric matrix equation
systemically. Author extended this work for several variable hypergeometric
matrix functions _viz._ Appell matrix functions, Lauricella matrix functions
of three and $n$ variables [1, 3, 4]. This motivates to study second order
matrix $q$-difference equation of bilateral type in a systematic way. Consider
the second order matrix $q$-difference equation of bilateral type

$\displaystyle\phi_{1,q}(z)\,D_{q}^{2}U(z)+D_{q}^{2}U(z)\,\phi_{2,q}(z)+\phi_{3,q}(z)\,D_{q}U(z)+D_{q}U(z)\,\phi_{4,q}(z)$
$\displaystyle\quad+\phi_{5,q}(z)\,D_{q}U(z)\,\phi_{6,q}(z)+\phi_{7,q}(z)\,U(z)\,\phi_{8,q}(z)=0,$
(3.3)

where $\phi_{i,q}(z),1\leq i\leq 8$ are bounded continuous functions in an
open disc $B(z_{0},p)$ of radius $p$ centred at $z_{0}$ given by
$\phi_{i,q}:B(z_{0},p)\rightarrow E_{q}$. Here $E_{q}$ is the $q$-Banach space
with $2$-norm such that $\lim_{q\rightarrow 1}E_{q}=E$ (Banach space of all
$p$-square matrices).

The point $z=z_{0}$ is called an ordinary point of $q$-difference equation
(3.3) if $\phi_{1,q}(z_{0})$ and $\phi_{2,q}(z_{0})$ are invertible. In case,
the inverse of $\phi_{1,q}(z_{0})$ and $\phi_{2,q}(z_{0})$ do not exist but
each $\phi_{j,q}(z_{0}),3\leq j\leq 8$ is invertible, then we called $z=z_{0}$
a singular point. Further, if limits

$\displaystyle\lim_{z\rightarrow
z_{0}}(z-z_{0})\,\phi_{j,q}(z)\,[\phi_{i,q}(z)]^{-1},\quad i=1,2,\ j=3,4,5,6$
(3.4)

and

$\displaystyle\lim_{z\rightarrow
z_{0}}(z-z_{0})^{2}\,\phi_{k,q}(z)\,[\phi_{i,q}(z)]^{-1},\quad i=1,2,\ k=7,8$
(3.5)

exist, then the point $z=z_{0}$ is called a regular singular point of (3.3).
If any of the limits given above do not exist, then the singular point is
irregular.

Let $\phi_{q}:B(z_{0};p)\times E_{q}\rightarrow E_{q}$ be a continuous
function such that

$\displaystyle\|\phi_{q}(z,X_{1,q})-\phi_{q}(z,X_{2,q})\|\leq
K\,(|z-z_{0}|)\,\|X_{1,q}-X_{2,q}\|,$ (3.6)

where $z\in B(z_{0},p)$; $X_{1,q},X_{2,q}\in E_{q}$ and $K_{q}$ is a real
valued continuous function on $[0,p]$. Then, there exists an unique solution
$W_{q}(z)$ of

$D_{q}(X_{q})=\phi_{q}(z,X_{q}),$ (3.7)

such that $W_{q}(z_{0})=X_{0,q}$.

Any two solutions $W_{1,q}$ and $W_{2,q}$ of second order $q$-difference
equation (3.3) are called fundamental set of solution if there exist a
solution $W_{q}$ of (3.3) which has a unique representation in the form

$\displaystyle W_{q}(z)=W_{1,q}(z)\,A+W_{2,q}(z)\,B,$ (3.8)

where matrices $A$ and $B$ are uniquely obtained by $W_{q}(z)$.

##  4 The $q$-Kummer matrix equation

Before discussing $q$-Kummer matrix equation and its solution we will define
$q$-Kummer matrix function and give its regions of convergence. Let $S$, $T$
be $p$-square matrices such that
$q^{-k}\notin\sigma(q^{T}),k\in\mathbb{Z}^{+}$. Then, we define the $q$-Kummer
matrix function as

$\displaystyle{}_{1}\phi_{1}(q^{S};q^{T};q;z)=\sum_{n\geq
0}\frac{(q^{S};q)_{n}(q^{T};q)_{n}^{-1}}{(q;q)_{n}}\,z^{n}.$ (4.1)

For convergence,

$\displaystyle\|{}_{1}\phi_{1}(q^{S};q^{T};q;z)\|$
$\displaystyle\leq\sum_{n\geq
0}\left\|\frac{(q^{S};q)_{n}(q^{T};q)_{n}^{-1}}{(q;q)_{n}}\,z^{n}\right\|$
$\displaystyle\leq\sum_{n\geq
0}\frac{(-\|q^{S}\|;|q|)_{n}\|(q^{T};q)_{n}^{-1}\|}{|(q;q)_{n}|}|z|^{n}=\sum_{n\geq
0}U_{n}\,|z|^{n}.$ (4.2)

Now, using the ratio test

$\displaystyle\lim_{n\rightarrow\infty}\left|\frac{U_{n+1}}{U_{n}}\right|$
$\displaystyle=\lim_{n\rightarrow\infty}\left|\frac{(-\|q^{S}\|;|q|)_{n+1}\|(q^{T};q)_{n+1}^{-1}\||z|^{n+1}|(q;q)_{n}|}{(-\|q^{S}\|;|q|)_{n}\|(q^{T};q)_{n}^{-1}\||z|^{n}|(q;q)_{n+1}|}\right|$
$\displaystyle=\lim_{n\rightarrow\infty}\left|\frac{(1+|q|^{n}\|q^{S}\|)\|(I-q^{n}q^{T})^{-1}\||z|}{1-|q|^{n+1}}\right|$
$\displaystyle\leq\lim_{n\rightarrow\infty}\frac{(1+|q|^{n}\|q^{S}\|)\,|z|}{(1-|q|^{n+1})\,(1-|q|^{n}\|q^{T}\|)}=|z|.$
(4.3)

Thus the $q$-Kummer matrix function ${}_{1}\phi_{1}(q^{S};q^{T};q;z)$
converges absolutely when $|q|<1$ and $|z|<1$.

Next, we give the matrix $q$-Kummer equation of bilateral type and discuss the
distinct solutions. Let $S$, $T$ be matrices in $\mathbb{C}^{p\times p}$ such
that $q^{-k}\notin\sigma(q^{T}),k\in\mathbb{Z}^{+}$. Then, we introduce the
matrix $q$-Kummer equation of bilateral type as

$\displaystyle
zD_{q}^{2}U(z)q^{T}+D_{q}U(z)[T]_{q}-q^{S}zD_{q}U(z)-[S]_{q}U(z)=0.$ (4.4)

This is an equation of the form (3.3), on comparing we have

$\displaystyle\phi_{2,q}(z)=z\,q^{T},$
$\displaystyle\quad\phi_{3,q}(z)=-z\,q^{S},\quad\phi_{4,q}(z)=[T]_{q},\quad\phi_{7,q}(z)=-[S]_{q}.$
(4.5)

The only singular points of the equation (4.4) are $z=0$ and $z=\infty$.
First, we give our attention in finding the series solution of the matrix
$q$-difference equation about the singular point $z=0$. Then we look into for
another form of solutions _viz._ solution at $\infty$ and integral solutions.
Let us assume the series solution of (4.4) be in the form

$\displaystyle U(z)=\sum_{n\geq 0}U_{n}z^{n}$ (4.6)

which yields

$\displaystyle D_{q}\,U(z)=\sum_{n\geq 1}U_{n}\,[n]_{q}\,z^{n-1},\quad
D_{q}^{2}\,U(z)=\sum_{n\geq 2}U_{n}\,[n]_{q}\,[n-1]_{q}\,z^{n-2}.$ (4.7)

Now, substituting these values in (4.4), we get

$\displaystyle\sum_{n\geq
2}U_{n}\,[n]_{q}\,[n-1]_{q}\,z^{n-1}q^{T}+\sum_{n\geq
1}U_{n}\,[n]_{q}\,z^{n-1}[T]_{q}-q^{S}\,\sum_{n\geq 1}U_{n}\,[n]_{q}\,z^{n}$
$\displaystyle\quad-[S]_{q}\sum_{n\geq 0}U_{n}\,z^{n}=0.$ (4.8)

After some manipulation, the equation (4.8) gives

$\displaystyle\sum_{n\geq
2}\left(U_{n+1}[n+1]_{q}\,[n]_{q}q^{T}+U_{n+1}[n+1]_{q}[T]_{q}-q^{S}U_{n}[n]_{q}-[S]_{q}U_{n}\right)\,z^{n}$
$\displaystyle\quad+(U_{2}[2]_{q}q^{T}+U_{2}[2]_{q}[T]_{q}-q^{S}U_{1}-[S]_{q}U_{1})\,z+U_{1}[T]_{q}-[S]_{q}U_{0}=0.$
(4.9)

Equating the powers of $z$ and hence one can obtain

$\displaystyle U_{1}\,[T]_{q}=[S]_{q}U_{0}\implies
U_{1}=\frac{[S]_{q}\,U_{0}\,[T]_{q}^{-1}\,(1-q)}{(1-q)};$ (4.10)
$\displaystyle U_{2}=\frac{[S+I]_{q}\,U_{1}\,[T+I]_{q}^{-1}}{[2]_{q}}$
$\displaystyle=\frac{[S+I]_{q}\,[S]_{q}\,U_{0}\,[T+I]_{q}^{-1}\,[T]_{q}^{-1}}{[2]_{q}}$
$\displaystyle=\frac{[S+I]_{q}\,[S]_{q}\,U_{0}\,[T+I]_{q}^{-1}\,[T]_{q}^{-1}\,(1-q)^{2}}{(1-q)\,(1-q^{2})}.$
(4.11)

In general

$\displaystyle U_{n+1}$
$\displaystyle=\frac{[S+nI]_{q}\,U_{n}\,[T+nI]_{q}^{-1}}{[n+1]_{q}}$
$\displaystyle=\frac{[S+nI]_{q}\,\cdots[S]_{q}\,U_{0}\,[T+nI]_{q}^{-1}\,\cdots[T]_{q}^{-1}\,(1-q)^{n}}{(1-q)\,(1-q^{2})\cdots(1-q^{n+1})}.$
(4.12)

Hence there exists a solution $U_{1}(z)$ of (4.4), well defined in
$|(1-q)\,z|<1$ and satisfying $U_{1}(0)=I$, is given by

$U_{1}(z)={}_{1}\phi_{1}(q^{S};q^{T};q;(1-q)\,z).$ (4.13)

Since equation (4.4) is a second order matrix difference equation, so it must
have another series solution. Suppose, for $ST=TS$, the second solution has
the form

$\displaystyle U(z)=V(z)\,z^{I-T},\quad z^{I-T}=\exp((I-T)\ln z).$ (4.14)

The $q$-derivatives of (4.14) are given by

$\displaystyle D_{q}U(z)$
$\displaystyle=V(z)\,[I-T]_{q}z^{-T}+D_{q}V(z)(qz)^{I-T}$ (4.15)
$\displaystyle D_{q}^{2}U(z)$
$\displaystyle=V(z)\,[I-T]_{q}\,[-T]_{q}\,z^{-T-I}+D_{q}V(z)\,[I-T]_{q}\,(qz)^{-T}+D_{q}^{2}V(z)(q^{2}z)^{I-T}$
$\displaystyle\quad+qD_{q}V(z)\,[I-T]_{q}\,(qz)^{-T}.$ (4.16)

Substituting these values in (4.4) yield

$\displaystyle
z\,D_{q}^{2}V(z)\,q^{2I-T}+D_{q}V(z)\,[2I-T]_{q}-z\,q^{S+I-T}\,D_{q}V(z)-[S+I-T]_{q}V(z)=0.$
(4.17)

The equation (4.17) is a matrix $q$-difference equation of type (4.4) and its
solution is given by ${}_{1}\phi_{1}(q^{\tilde{S}};q^{\tilde{T}};q;(1-q)\,z)$,
where $\tilde{S}=S+I-T$ and $\tilde{T}=2I-T$. Thus we can conclude that the
matrix $q$-Kummer equation (4.4) has two fundamental solutions as

$\displaystyle U_{1}(z)$
$\displaystyle={}_{1}\phi_{1}(q^{S};q^{T};q;(1-q)\,z);$ $\displaystyle
U_{2}(z)$
$\displaystyle=z^{I-T}\,{}_{1}\phi_{1}(q^{S+I-T};q^{2I-T};q;(1-q)\,z),\quad
ST=TS.$ (4.18)

In view of the convergence condition for $q$-Kummer matrix function, the
solutions $U_{1}(z)$ and $U_{2}(z)$ converges absolutely in the region $|q|<1$
and $|(1-q)\,z|<1$.

###  4.1 Solutions at $\infty$

Now, we seek for the solution of matrix $q$-Kummer equation at infinity and so
we rewrite the equation (4.4) as

$\displaystyle
U(q^{2}z)q^{T-I}-[I+q^{S}(q-1)z]\,U(qz)-U(qz)\,q^{T-I}+[1+(q-1)z]\,U(z)=0.$
(4.19)

Let $u=z^{-1}$, $s=q^{-1}$, $W(u)=U(z^{-1})$ and $ST=TS$. Then (4.19) becomes

$\displaystyle u\,W(s^{2}u)-[(1-s)\,s^{T-S-2I}+(I+s^{T-I})\,u]\,W(su)$
$\displaystyle\quad+[(1-s)\,s^{T-2I}+u\,s^{T-I}]\,W(u)=0.$ (4.20)

Which can be written in the derivative form as

$\displaystyle
u^{3}\,D_{s}^{2}W(u)+[u\,s^{T-S-3I}-u^{2}\,[T-2I]_{s}]D_{s}\,W(u)+s^{T-3I}\,[-S]_{s}\,W(u)=0,$
(4.21)

where $D_{s}\,W(u)=\frac{W(u)-W(su)}{(1-s)\,u}$ and
$[S]_{s}=\frac{I-s^{S}}{1-s}$. This is the matrix $s$-difference equation of
the form (3.3) where

$\displaystyle\phi_{1,s}(u)=u^{3},\
\phi_{3,s}(u)=u\,s^{T-S-3I}-u^{2}\,[T-2I]_{s},\
\phi_{7,s}(u)=s^{T-3I}\,[-S]_{s}.$ (4.22)

All of the above are well defined near $u=0$ except the limit

$\displaystyle\lim_{u\rightarrow
0}\frac{u\,\phi_{3,s}(u)}{\phi_{1,s}(u)}=\lim_{u\rightarrow
0}\frac{u\,[u\,s^{T-S-3I}-u^{2}\,[T-2I]_{s}]}{u^{3}}.$ (4.23)

Thus, point $u=0$ is irregular singular for (4.21) and hence the point
$z=\infty$ is irregular for (4.4). Therefore, the equation (4.4) does not have
a solution at $\infty$. Unfortunately, we are not getting a convergent
solution of (4.4) at $z=\infty$ but it is possible to get some integral
solutions which are being discussed in the next section.

###  4.2 Integral solutions of $q$-Kummer matrix equation

Consider one of the integral solution of (4.4) in the form

$\displaystyle
U_{1}(z)=\int_{0}^{\frac{1}{(1-q)q^{2}z}}E_{q}^{-qzu}\,f(qu)\,d_{q}u,$ (4.24)

where $E_{q}^{-qzu}$ and its $q$-derivative are defined by [7]

$\displaystyle E_{q}^{-qzu}=((1-q)qzu;q)_{\infty}\text{ and
}D_{q}E_{q}^{-qzu}=-qu\,E_{q}^{-q^{2}zu}.$ (4.25)

Using (2.18) and (4.25), one can get

$\displaystyle D_{q}\,U_{1}(z)$
$\displaystyle=\int_{0}^{\frac{1}{(1-q)q^{2}z}}(-qu)\,E_{q}^{-q^{2}zu}\,f(qu)\,d_{q}u-\left(\frac{1}{(1-q)q^{3}z^{2}}\,E_{q}^{-q^{2}zu}f(qu)\right)_{u=1/(1-q)q^{3}z}$
$\displaystyle=-\int_{0}^{\frac{1}{(1-q)q^{2}z}}qu\,E_{q}^{-q^{2}zu}\,f(qu)\,d_{q}u,$
(4.26)

and

$\displaystyle D_{q}^{2}\,U_{1}(z)$
$\displaystyle=-D_{q}\left(\int_{0}^{\frac{1}{(1-q)q^{2}z}}qu\,E_{q}^{-q^{2}zu}\,f(qu)\,d_{q}u\right)$
$\displaystyle=\int_{0}^{\frac{1}{(1-q)q^{2}z}}q^{3}u^{2}\,E_{q}^{-q^{3}zu}\,f(qu)\,d_{q}u.$
(4.27)

Substituting these values in (4.4), we get

$\displaystyle\int_{0}^{\frac{1}{(1-q)q^{2}z}}[-f(qu)D_{q,u}(E_{q}^{-q^{2}zu})q^{T+I}u^{2}-u\,q^{S}\,f(qu)\,D_{q,u}(E_{q}^{-qzu})$
$\displaystyle\quad-
quf(qu)E_{q}^{-q^{2}zu}[T]_{q}-[S]_{q}f(qu)E_{q}^{-qzu}]d_{q}u$
$\displaystyle=\int_{0}^{\frac{1}{(1-q)q^{2}z}}E_{q}^{-q^{2}zu}\,D_{q,u}(f(u)q^{T-I}u^{2})+E_{q}^{-qzu}D_{q,u}(q^{S-I}f(u)\,u)$
$\displaystyle\quad-
quf(qu)E_{q}^{-q^{2}zu}[T]_{q}-[S]_{q}f(qu)E_{q}^{-qzu}]d_{q}u$
$\displaystyle=\int_{0}^{\frac{1}{(1-q)q^{2}z}}E_{q}^{-q^{2}zu}\\{D_{q,u}(f(u)q^{T-I}u^{2})+(1-(1-q)qzu)D_{q,u}(q^{S-I}f(u)\,u)$
$\displaystyle\quad-
quf(qu)[T]_{q}-[S]_{q}f(qu)+(1-q)qzu[S]_{q}f(qu)\\}d_{q}u=0,$ (4.28)

where $D_{q,u}$ is the $q$-derivative with respect to $u$. Since the integral
value is zero so one of the possibility is that

$\displaystyle
D_{q,u}(f(u)q^{T-I}u^{2})+(1-(1-q)qzu)D_{q,u}(q^{S-I}f(u)\,u)-quf(qu)[T]_{q}$
$\displaystyle\quad-[S]_{q}f(qu)+(1-q)qzu[S]_{q}f(qu)=0.$ (4.29)

Expanding the term containing $q$-derivative, we get a difference equation in
terms of $f(u)$ as

$\displaystyle\\{1+[q-(1-q)qz]\,u\\}f(qu)-q^{S-I}\\{1-(1-q)qzu\\}f(u)-uf(u)q^{T-I}=0.$
(4.30)

One can easily verify that for $ST=TS$ the matrix function,

$\displaystyle
F(u)=u^{S-I}\,([(1-q)qz-q]u;q)_{\infty}\,([(1-q)qz-q^{T-S}]u;q)_{\infty}^{-1},$
(4.31)

is a solution of recurrence relation (4.30). Thus $U_{1}(z)$ can be expressed
as

$\displaystyle U_{1}(z)$
$\displaystyle=\int_{0}^{\frac{1}{(1-q)q^{2}z}}E_{q}^{-qzu}\,{(qu)}^{S-I}([(1-q)qz-q]qu;q)_{\infty}$
$\displaystyle\quad\times([(1-q)qz-q^{T-S}]qu;q)_{\infty}^{-1}\,d_{q}u.$
(4.32)

This is an integral solution of $q$-Kummer matrix equation (4.4). We already
discussed, a second order difference equation must have two fundamental
solutions. So there is an opportunity of getting another integral solution. To
get the second solution we use the another $q$-analogue of exponential
function exists in the literature [7]. We will look the second solution in the
form

$\displaystyle U_{2}(z)=\int_{0}^{\infty}e_{q}^{-zu}\,f(qu)\,d_{q}u,$ (4.33)

where

$\displaystyle e_{q}^{-zu}=\frac{1}{(-(1-q)zu;q)_{\infty}}.$ (4.34)

The $q$-derivative of $e_{q}^{-zu}$ with respect to $z$ is

$\displaystyle D_{q}e_{q}^{-zu}=-u\,e_{q}^{-zu}.$ (4.35)

Further, using (2.18) and (4.35), one can get

$\displaystyle D_{q}U_{2}(z)=-\int_{0}^{\infty}u\,e_{q}^{-zu}\,f(qu)\,d_{q}u,$
(4.36) $\displaystyle
D_{q}^{2}U_{2}(z)=\int_{0}^{\infty}u^{2}\,e_{q}^{-zu}\,f(qu)\,d_{q}u.$ (4.37)

Substituting these values in (4.4), we have

$\displaystyle\int_{0}^{\infty}f(qu)[-(D_{q,u}e_{q}^{-zu})q^{T}u^{2}-(D_{q,u}e_{q}^{-zu})q^{S}u-([S]_{q}+[T]_{q}u)e_{q}^{-zu}]d_{q}u=0.$
(4.38)

Which can be rewritten as

$\displaystyle\int_{0}^{\infty}e_{q}^{-zu}[D_{q,u}(f(u)q^{T-2I}u^{2})+D_{q,u}(f(u)q^{S-I}u)-f(qu)([S]_{q}+[T]_{q}u)]d_{q}u=0.$
(4.39)

From above equation, one can obtain a matrix $q$-difference equation in terms
of $f(u)$ as

$\displaystyle(1+u)f(qu)-(q^{S-I}+q^{T-2I}u)f(u)=0.$ (4.40)

It is easy to verify that the matrix $q$-difference equation (4.40) has
solution

$\displaystyle f(u)=u^{S-I}{(-u;q)_{\infty}}(-q^{S-T-I}u;q)_{\infty}^{-1}.$
(4.41)

Thus, $U_{2}(z)$ can be expressed as

$\displaystyle
U_{2}(z)=q^{S-I}\,\int_{0}^{\infty}u^{S-I}\,e_{q}^{-zu}\,{(-qu;q)_{\infty}}(-q^{S-T}u;q)_{\infty}^{-1}\,d_{q}u.$
(4.42)

We now summarize the integral solutions of (4.4) in the following theorem.

######  Theorem 4.1.

Let $S$ and $T$ be commuting matrices in $\mathbb{C}^{p\times p}$ such that
$S$, $T$ and $T-S$ are positive stable. Then the matrix equation (4.4) has two
integral solutions as

$\displaystyle U_{1}(z)$
$\displaystyle=\int_{0}^{\frac{1}{(1-q)q^{2}z}}E_{q}^{-qzu}\,{(qu)}^{S-I}([(1-q)qz-q]qu;q)_{\infty}$
$\displaystyle\quad\times([(1-q)qz-q^{T-S}]qu;q)_{\infty}^{-1}\,d_{q}u;$
$\displaystyle U_{2}(z)$
$\displaystyle=q^{S-I}\,\int_{0}^{\infty}u^{S-I}\,e_{q}^{-zu}\,{(-qu;q)_{\infty}}(-q^{S-T}u;q)_{\infty}^{-1}\,d_{q}u.$
(4.43)

##  5 Conclusion

In this paper, we determined the solutions of $q$-Kummer matrix equation of
bilateral type. We discussed the series solutions, solution at infinity and
integral solutions. One can observe that the only solution for non-commutative
$S$ and $T$ is in (4.13). For other solutions, we are directly or indirectly
assuming that $S$ and $T$ are commuting.

The present work of ${}_{1}\phi_{1}(q^{S};q^{T};q;(1-q)z)$ can be easily
extended for several variable basic hypergeometric matrix function. One can
introduce the matrix $q$-difference equation of bilateral type and obtain its
distinct form of solutions. In particular, let $P$, $Q$, $R$ be matrices in
$\mathbb{C}^{p\times p}$ such that
$q^{-k}\notin\sigma(q^{R}),k\in\mathbb{Z}^{+}$. Then, it is easy to show that
the matrix $q$-difference equation,

$\displaystyle\left[\delta_{q}(q^{R-I}\,\delta_{q}+[R-I]_{q})-z(q^{P}\,\delta_{q}+[P]_{q})\,(q^{Q}\,\delta_{q}+[Q]_{q})\right]W(z)=0,$
(5.1)

has basic Gauss hypergeometric matrix function
${}_{2}\phi_{1}(q^{P},q^{Q};q^{R};q;z)$ as a solution provided the matrices
$P$, $Q$, $R$ commutes with each other. To overcome with such comprehensive
assumption of commutativity, one can introduce a second order bilateral type
matrix $q$-difference equation

$\displaystyle
zD_{q}^{2}\,W(z)\,q^{R}-z[P]_{q}\,D_{q}W(z)\,q^{Q}+D_{q}W(z)\,[R]_{q}-z\,q^{P}\,D_{q}W(z)[Q+I]_{q}$
$\displaystyle\quad-z^{2}q^{P+I}D_{q}^{2}W(z)\,q^{Q}-[P]_{q}\,W(z)\,[Q]_{q}=0.$
(5.2)

The equation (5.2) is a $q$-analogue of matrix Gauss hypergeometric equation
given in [10]. On comparing the above equation with (3.3), we get

$\displaystyle\phi_{1,q}(z)$ $\displaystyle=-z^{2}\,q^{P+I},\
\phi_{2,q}(z)=z\,q^{R-Q},\ \phi_{3,q}(z)=-z\,[P]_{q},\
\phi_{4,q}(z)=[R]_{q}\,q^{-Q},$ $\displaystyle\phi_{5,q}(z)$
$\displaystyle=-z\,q^{P},\ \phi_{6,q}(z)=[Q+I]_{q}\,q^{-Q},\
\phi_{7,q}(z)=-[P]_{q},\ \phi_{8,q}(z)=[Q]_{q}\,q^{-Q}.$ (5.3)

Treating in the same way to matrix $q$-Kummer equation, one can obtain that
the series solutions of the equation (5.2) are

$\displaystyle W_{1}(z)$
$\displaystyle={}_{2}\phi_{1}(q^{P},q^{Q};q^{R};q;z),\quad QR=RQ;$
$\displaystyle W_{2}(z)$
$\displaystyle=z^{I-R}\,{}_{2}\phi_{1}(q^{P+I-R},q^{Q+I-R};q^{2I-R};q;z),\quad
PR=RP,QR=RQ.$ (5.4)

Finding the other form of solutions of matrix difference equation (5.2) is an
interesting problem and yet to be done. To find the matrix $q$-difference
equation satisfied by basic Gauss hypergeometric matrix function for non-
commutative matrices is also an open problem and may be obtained in my
upcoming articles.

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