# Hypergeometric Expressions for Type I Jacobi–Piñeiro Orthogonal Polynomials
with Arbitrary Number of Weights

Amílcar Branquinho1 1CMUC, Departamento de Matemática, Universidade de
Coimbra, Largo D. Dinis, 3000-143 Coimbra, Portugal<EMAIL_ADDRESS>,  Juan
E. F. Díaz2 2CIDMA, Departamento de Matemática, Universidade de Aveiro,
3810-193 Aveiro, Portugal<EMAIL_ADDRESS>,  Ana Foulquié-Moreno3 3CIDMA,
Departamento de Matemática, Universidade de Aveiro, 3810-193 Aveiro, Portugal
<EMAIL_ADDRESS>and  Manuel Mañas4 4Departamento de Física Teórica,
Universidad Complutense de Madrid, 28040-Madrid, Spain<EMAIL_ADDRESS>
###### Abstract.

For a general number $p\geq 2$ of measures, we provide explicit expressions
for the Jacobi–Piñeiro and Laguerre of the first kind multiple orthogonal
polynomials of type I, presented in terms of multiple hypergeometric
functions.

###### Key words and phrases:

Multiple orthogonal polynomials, hypergeometric series, Kampé de Fériet
series, Jacobi–Piñeiro, Laguerre, AT systems

###### 2020 Mathematics Subject Classification:

33C45, 33C47, 42C05, 47A56.

1Acknowledges Centro de Matemática da Universidade de Coimbra (CMUC) –
UIDB/00324/2020, funded by the Portuguese Government through FCT/MEC and co-
funded by the European Regional Development Fund through the Partnership
Agreement PT2020.

2 and 3Acknowledges CIDMA Center for Research and Development in Mathematics
and Applications (University of Aveiro) and the Portuguese Foundation for
Science and Technology (FCT) within project UIDB/04106/2020 and
UIDP/04106/2020.

The authors acknowledges “Agencia Estatal de Investigación” research project
[PID2021- 122154NB-I00], _Ortogonalidad y aproximación con aplicaciones en
machine learning y teoría de la probabilidad_.

##  1\. Introduction

Multiple orthogonal polynomials are a specialized family of polynomials that
arise in various branches of mathematics and engineering. Unlike classical
orthogonal polynomials, which are associated with a single weight function,
multiple orthogonal polynomials are linked to multiple weight functions and
measures simultaneously. These polynomials play a crucial role in diverse
fields, including numerical analysis, approximation theory, and mathematical
physics, offering powerful tools for solving complex problems involving
simultaneous orthogonalities. In the book [16] there is a comprehensive
introduction to the subject and [1] for the relation with integrable systems.

Recently, the importance of multiple orthogonal polynomials in the Favard
spectral description of banded bounded semi-infinite matrices has come to
light. This connection has been explored in various sources such as [12, 14,
15], see also [13]. Additionally, their significance in the context of Markov
chains and random walks beyond birth and death has been established, as seen
in [11, 10, 9, 8]. Notably, in both of these scenarios, type I polynomials
play a pivotal role. Unfortunately, explicit expressions for type I multiple
orthogonal polynomials are not widely available. In contrast, for type II
polynomials, the Rodrigues’ formula leads to explicit hypergeometric
expressions that can accommodate an arbitrary number of ‘classical’ weights,
as detailed in [5] and [3, 4, 21].

In our work presented in [6], we offer explicit hypergeometric expressions for
all type I polynomials within the realm of multiple Hahn orthogonal
polynomials under the Askey scheme [5, 18], a context encompassing only two
weights. Through these efforts, we have successfully constructed bidiagonal
factorizations and characterized positive ones [7], as elaborated in [12, 14].

In this paper, we provide explicit hypergeometric expressions for the type I
Jacobi–Piñeiro and type I Laguerre multiple orthogonal polynomials of the
first kind, for the first time. To achieve this, we relied on tools acquired
from our previous work in [6] and offered a fresh interpretation of results
presented by Karp and Prilepkina in [17].

For the upcoming discussion, it is essential to establish some fundamental
concepts about multiple orthogonal polynomials [19]. We begin by examining a
system characterized by $p\geq 2$ weight functions, denoted as
$w_{1},\ldots,w_{p}$, which map from $\Delta\subseteq\mathbb{R}$ to
$\mathbb{R}$, in conjunction with a measure
$\mu:\Delta\subseteq\mathbb{R}\rightarrow\mathbb{R}^{+}$.

In certain cases, a sequence of type II polynomials $B_{\vec{n}}$ may exist,
which are monic with $\deg{B}\leq|\vec{n}|$, satisfying the orthogonality
relations

(1)
$\displaystyle\int_{\Delta}x^{j}B_{\vec{n}}(x)w_{i}(x)\operatorname{d}\mu(x)=0,$
$\displaystyle j\in\\{0,\dots,n_{i}-1\\},$ $\displaystyle
i\in\\{1,\dots,p\\}.$

Additionally, there are $p$ sequences of type I polynomials, denoted as
$A^{(1)}_{\vec{n}},\ldots,A^{(p)}_{\vec{n}}$, with $\deg A^{(i)}_{\vec{n}}\leq
n_{i}-1$, meeting the orthogonality conditions

(2)
$\displaystyle\sum_{i=1}^{p}\int_{\Delta}x^{j}A^{(i)}_{\vec{n}}(x)w_{i}(x)\operatorname{d}\mu(x)=\begin{cases}0,\;\text{if}\;j\in\\{0,\ldots,|\vec{n}|-2\\},\\\
1,\;\text{if}\;j=|\vec{n}|-1.\end{cases}$

In this context, $\vec{n}=(n_{1},\ldots,n_{p})\in\mathbb{N}^{p}_{0}$ and
$|\vec{n}|\coloneqq n_{1}+\cdots+n_{p}$. In this paper we will use the
notation $\mathbb{N}\coloneq\\{1,2,3,\dots\\}$ and
$\mathbb{N}_{0}\coloneq\\{0\\}\cup\mathbb{N}$.

Before delving into the main results, it’s important to recall that, in the
context of standard orthogonality and for certain families when $p=2$, the
polynomials can be represented using generalized hypergeometric series, as
discussed in [2]. These series, denoted as

(3) $\displaystyle{}_{p}F_{q}\biggl{[}\genfrac{.}{.}{0.0pt}{}{a_{1}\mskip
8.0mu\ldots\mskip 8.0mua_{p}}{\alpha_{1}\mskip 8.0mu\ldots\mskip
8.0mu\alpha_{q}};x\biggr{]}\coloneqq\sum_{l=0}^{\infty}\dfrac{(a_{1})_{l}\cdots(a_{p})_{l}}{(\alpha_{1})_{l}\cdots(\alpha_{q})_{l}}\dfrac{x^{l}}{l!}.$

Here, $(x)_{n}$, $x\in\mathbb{C}$ and $n\in\mathbb{N}_{0}$, known as the
Pochhammer symbol, is defined as:

$\displaystyle(x)_{n}\coloneqq\dfrac{\Gamma(x+n)}{\Gamma(x)}=\begin{cases}x(x+1)\cdots(x+n-1),\;\text{if}\;n\in\mathbb{N},\\\
1,\;\text{if}\;n=0.\end{cases}$

To establish the results related to the type I polynomials, we will rely on
the theorem by Karp and Prilepkina, as presented in [17, Theorem 2.2].

######  Theorem 1.

Let be
$a\in\mathbb{C},\,\vec{f}=(f_{1},\ldots,f_{r})\in\mathbb{C}^{r},\,\vec{b}=(b_{1},\ldots,b_{l})\in\mathbb{C}^{l}$,
$\vec{m}=(m_{1},\ldots,$ $m_{r})\in\mathbb{N}^{r}$ and
$\vec{k}=(k_{1},\ldots,k_{l})\in\mathbb{N}^{l}$ such that all the elements of
the vector

$(\beta_{1},\beta_{2},\ldots,\beta_{k_{1}},\ldots,\beta_{k_{1}+\cdots+k_{l-1}+1},\beta_{k_{1}+\cdots+k_{l-1}+2},\ldots,\beta_{k_{1}+\cdots+k_{l}})\\\
\coloneqq(b_{1},b_{1}+1,\ldots,b_{1}+k_{1}-1,\ldots,b_{l},b_{l}+1,\ldots,b_{l}+k_{l}-1)$

are distinct. If $\operatorname{Re}(k_{1}+\cdots+k_{l}-a-m_{1}-\cdots-
m_{r})>0$, then

(4) ${}_{r+l+1}F_{r+l}\biggl{[}\genfrac{.}{.}{0.0pt}{}{a\mskip
8.0muf_{1}+m_{1}\mskip 8.0mu\ldots\mskip 8.0muf_{r}+m_{r}\mskip
8.0mub_{1}\mskip 8.0mu\ldots\mskip 8.0mub_{l}}{f_{1}\mskip 8.0mu\ldots\mskip
8.0muf_{r}\mskip 8.0mub_{1}+k_{1}\mskip 8.0mu\ldots\mskip
8.0mub_{l}+k_{l}};1\biggr{]}\\\
=\Gamma(1-a)\dfrac{(b_{1})_{k_{1}}\cdots(b_{l})_{k_{l}}}{(f_{1})_{m_{1}}\cdots(f_{r})_{m_{r}}}\sum_{i=1}^{k_{1}+\cdots+k_{l}}\dfrac{\Gamma(\beta_{i})}{\Gamma(\beta_{i}-a+1)}\dfrac{\prod_{j=1}^{r}(f_{j}-\beta_{i})_{m_{j}}}{\tensor*{\prod\nolimits}{{}^{k_{1}+\cdots+k_{l}}_{j=1}{}^{,(i)}}(\beta_{j}-\beta_{i})}.$

######  Remark 1.

Notice that here we have introduced the following notation

$\displaystyle{\prod}^{p,(i)}_{j=1}a_{j},$ $\displaystyle
i\in\\{1,\ldots,p\\},$

to indicate that the $i$-th factor is removed from the product. In the
forthcoming discussion, we will make extensive use of this as well as the
following notation:

$\displaystyle\vec{e}_{p}\coloneqq(1,\ldots,1)\in\mathbb{R}^{p}.$

Given a vector $\vec{a}\in\mathbb{R}^{p}$, the notation
$\vec{a}^{(i)}\in\mathbb{R}^{p-1}$ indicates that the $i$-th component of
$\vec{a}$ has been removed.

However, in this context, we will rephrase this theorem into an equivalent,
more practical form

######  Lemma 1.

Let be
$a\in\mathbb{C},\,\vec{f}=(f_{1},\ldots,f_{r})\in\mathbb{C}^{r},\,\vec{b}=(b_{1},\ldots,b_{l})\in\mathbb{C}^{l}$
with all its components distinct,
$\vec{m}=(m_{1},\ldots,m_{r})\in\mathbb{N}^{r}$ and
$\vec{k}=(k_{1},\ldots,k_{l})\in\mathbb{N}^{l}$. If
$\operatorname{Re}(k_{1}+\cdots+k_{l}-a-m_{1}-\cdots-m_{r})>0$, then

(5) ${}_{r+l+1}F_{r+l}\biggl{[}\genfrac{.}{.}{0.0pt}{}{a\mskip
8.0muf_{1}+m_{1}\mskip 8.0mu\ldots\mskip 8.0muf_{r}+m_{r}\mskip
8.0mub_{1}\mskip 8.0mu\ldots\mskip 8.0mub_{l}}{f_{1}\mskip 8.0mu\ldots\mskip
8.0muf_{r}\mskip 8.0mub_{1}+k_{1}\mskip 8.0mu\ldots\mskip
8.0mub_{l}+k_{l}};1\biggr{]}\\\
=\Gamma(1-a)\dfrac{(b_{1})_{k_{1}}\cdots(b_{l})_{k_{l}}}{(f_{1})_{m_{1}}\cdots(f_{r})_{m_{r}}}\sum_{q=1}^{l}\dfrac{(-1)^{k_{q}-1}\prod_{j=1}^{r}(\tilde{f}_{j}-m_{j})_{m_{j}}\Gamma(b_{q}+k_{q}-1)}{(k_{q}-1)!\prod^{l,(q)}_{j=1}(b_{j}-b_{q}-k_{q}+1)_{k_{j}}\Gamma(b_{q}+k_{q}-a)}\\\
\times{}_{r+l+1}F_{r+l}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-k_{q}+1\mskip 8.0mu-
b_{q}-k_{q}+1+a\mskip 8.0mu\tilde{f}_{1}\mskip 8.0mu\ldots\mskip
8.0mu\tilde{f}_{r}\mskip
8.0mu\vec{b}^{(q)}-(b_{q}+k_{q}-1)\vec{e}_{l-1}}{-b_{q}-k_{q}+2\mskip
8.0mu\tilde{f}_{1}-m_{1}\mskip 8.0mu\ldots\mskip
8.0mu\tilde{f}_{r}-m_{r}\mskip
8.0mu\,\vec{b}^{(q)}+\vec{k}^{(q)}-(b_{q}+k_{q}-1)\vec{e}_{l-1}};1\biggr{]},$

with $\tilde{f}_{j}=f_{j}-b_{q}+1-k_{q}+m_{j}$, $j=1,\ldots,r$.

###### Proof.

For the sake of clarity, we’ll temporarily assign a new name to the expression
within the sum in Equation (4) as follows:

$\displaystyle
F(\beta_{i})\coloneqq\dfrac{\Gamma(\beta_{i})}{\Gamma(\beta_{i}-a+1)}\dfrac{\prod_{j=1}^{r}(f_{j}-\beta_{i})_{m_{j}}}{\prod_{j=1}^{k_{1}+\cdots+k_{l},(i)}(\beta_{j}-\beta_{i})}.$

Now, we can reorganize this sum as follows:

$\displaystyle\sum_{i=1}^{k_{1}+\cdots+k_{l}}F(\beta_{i})$
$\displaystyle=\sum_{i=1}^{k_{1}}F(\beta_{i})+\sum_{i=k_{1}+1}^{k_{1}+k_{2}}F(\beta_{i})+\cdots+\sum_{i=k_{1}+\cdots+k_{l-1}+1}^{k_{1}+\cdots+k_{l}}F(\beta_{i})$
$\displaystyle=\sum_{i=1}^{k_{1}}F(b_{1}+i-1)+\sum_{i=1}^{k_{2}}F(b_{2}+i-1)+\cdots+\sum_{i=1}^{k_{l}}F(b_{l}+i-1)$
$\displaystyle=\sum_{q=1}^{l}\sum_{i=1}^{k_{q}}F(b_{q}+i-1).$

Replacing $F(b_{q}+i-1)$ and simplifying, we can rewrite each $i$-labeled sum
as a ${}_{r+l+1}F_{r+l}$ function, resulting in the following expression:

${}_{r+l+1}F_{r+l}\biggl{[}\genfrac{.}{.}{0.0pt}{}{a\mskip
8.0muf_{1}+m_{1}\mskip 8.0mu\ldots\mskip 8.0muf_{r}+m_{r}\mskip
8.0mub_{1}\mskip 8.0mu\ldots\mskip 8.0mub_{l}}{f_{1}\mskip 8.0mu\ldots\mskip
8.0muf_{r}\mskip 8.0mub_{1}+k_{1}\mskip 8.0mu\ldots\mskip
8.0mub_{l}+k_{l}};1\biggr{]}=\Gamma(1-a)\dfrac{(b_{1})_{k_{1}}\cdots(b_{l})_{k_{l}}}{(f_{1})_{m_{1}}\cdots(f_{r})_{m_{r}}}\\\
\times\sum_{q=1}^{l}\dfrac{\prod_{j=1}^{r}(f_{j}-b_{q})_{m_{j}}\Gamma(b_{q})}{(k_{q}-1)!\prod_{j=1}^{l,(q)}(b_{j}-b_{q})_{k_{j}}\Gamma(b_{q}+1-a)}\\\
\times{}_{r+l+1}F_{r+l}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-k_{q}+1\mskip
8.0mub_{q}\mskip 8.0mu-f_{1}+b_{q}+1\mskip 8.0mu\ldots\mskip 8.0mu-
f_{r}+b_{q}+1\mskip
8.0mu\,(b_{q}+1)\vec{e}_{l-1}-\vec{b}^{(q)}-\vec{k}^{(q)}}{b_{q}+1-a\mskip
8.0mu-f_{1}+b_{q}-m_{1}+1\mskip 8.0mu\ldots\mskip 8.0mu-
f_{r}+b_{q}-m_{r}+1\mskip
8.0mu\,(b_{q}+1)\vec{e}_{l-1}-\vec{b}^{(q)}};1\biggr{]}.$

Finally, applying the following reversal formula, as seen in [20, Equation
2.2.3.2]:

${}_{p+1}F_{q}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-n\mskip 8.0mua_{1}\mskip
8.0mu\ldots\mskip 8.0mua_{p}}{b_{1}\mskip 8.0mu\ldots\mskip
8.0mub_{q}};1\biggr{]}\\\
=(-1)^{n}\dfrac{(a_{1})_{n}\cdots(a_{p})_{n}}{(b_{1})_{n}\cdots(b_{q})_{n}}{}_{q+1}F_{p}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-n\mskip
8.0mu-b_{1}-n+1\mskip 8.0mu\ldots\mskip 8.0mu-b_{q}-n+1}{-a_{1}-n+1\mskip
8.0mu\ldots\mskip 8.0mu-a_{p}-n+1};1\biggr{]}.$

We can apply this reversal formula to each of the ${}_{r+l+1}F_{r+l}$
functions within the $q$-labeled sum and simplify to obtain expression (5). ∎

##  2\. Hypergeometric expressions for type I Jacobi–Piñeiro multiple
orthogonal polynomials

For this family, the weight functions are defined as follows:

(6) $\displaystyle w_{i}(x;\alpha_{i})=x^{\alpha_{i}},$ $\displaystyle
i\in\\{1,\ldots,p\\},$
$\displaystyle\operatorname{d}\mu(x)=(1-x)^{\beta}\operatorname{d}x,$
$\displaystyle\Delta=[0,1].$

Here $\vec{\alpha}=(\alpha_{1},\ldots,\alpha_{p})$ with
$\alpha_{1},\ldots,\alpha_{p},\beta>-1$, and, to establish an AT system, we
require that $\alpha_{i}-\alpha_{j}\not\in\mathbb{Z}$ for $i\neq j$. It’s also
important to note that the moments are given by:

(7)
$\displaystyle\int_{0}^{1}x^{\alpha_{i}+k}(1-x)^{\beta}\operatorname{d}x=\dfrac{\Gamma(\beta+1)\Gamma(\alpha_{i}+k+1)}{\Gamma(\alpha_{i}+\beta+k+2)},$
$\displaystyle k\in\mathbb{N}_{0}.$

In [6, §3] and [11], the type I polynomials were derived for the case when
$p=2$ and expressed using a ${}_{3}F_{2}$ function as follows:

$P^{(i)}_{(n_{1},n_{2})}(x;\alpha_{1},\alpha_{2},\beta)=(-1)^{n_{1}+n_{2}-1}\dfrac{(\alpha_{1}+\beta+n_{1}+n_{2})_{n_{1}}({\alpha}_{2}+\beta+n_{1}+n_{2})_{{n}_{2}}}{(n_{i}-1)!(\hat{\alpha}_{i}-\alpha_{i})_{\hat{n}_{i}}}\\\
\times\dfrac{\Gamma(\alpha_{i}+\beta+n_{1}+n_{2})}{\Gamma(\beta+n_{1}+n_{2})\Gamma(\alpha_{i}+1)}{}_{3}F_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-n_{i}+1\mskip
8.0mu\alpha_{i}+\beta+n_{1}+n_{2}\mskip
8.0mu\alpha_{i}-\hat{\alpha}_{i}-\hat{n}_{i}+1}{\alpha_{i}+1\mskip
8.0mu\alpha_{i}-\hat{\alpha}_{i}+1};x\biggr{]}.$

Here,
$\hat{\alpha}_{i}\coloneqq\delta_{i,2}\alpha_{1}+\delta_{i,1}\alpha_{2}$,
$\hat{n}_{i}\coloneqq\delta_{i,2}n_{1}+\delta_{i,1}n_{2}$ with
$i\in\\{1,2\\}$.

In an attempt to generalize this expression, we arrive at the following
theorem:

######  Theorem 2.

The Jacobi–Piñeiro polynomials of type I for $p\geq 2$ are given by:

(8)
$P^{(i)}_{\vec{n}}(x;\alpha_{1},\ldots,\alpha_{p},\beta)=(-1)^{|\vec{n}|-1}\dfrac{\prod_{k=1}^{p}(\alpha_{k}+\beta+|\vec{n}|)_{n_{k}}}{(n_{i}-1)!\prod_{k=1}^{p,(i)}(\alpha_{k}-\alpha_{i})_{n_{k}}}\\\
\times\dfrac{\Gamma(\alpha_{i}+\beta+|\vec{n}|)}{\Gamma(\beta+|\vec{n}|)\Gamma(\alpha_{i}+1)}{}_{p+1}F_{p}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-n_{i}+1\mskip
8.0mu\alpha_{i}+\beta+|\vec{n}|\mskip
8.0mu(\alpha_{i}+1)\vec{e}_{p-1}-\vec{\alpha}^{(i)}-\vec{n}^{(i)}}{\alpha_{i}+1\mskip
8.0mu(\alpha_{i}+1)\vec{e}_{p-1}-\vec{\alpha}^{(i)}};x\biggr{]}.$

###### Proof.

Initially, we will leverage Equation (7) and our understanding of
hypergeometric functions to derive a more convenient expression for the
following integral:

$\int_{0}^{1}x^{j}P^{(i)}_{\vec{n}}(x;\alpha_{1},\ldots,\alpha_{p},\beta)x^{\alpha_{i}}(1-x)^{\beta}\operatorname{d}x\\\
\begin{aligned}
&=\begin{multlined}(-1)^{|\vec{n}|-1}\dfrac{\prod_{k=1}^{p}(\alpha_{k}+\beta+|\vec{n}|)_{n_{k}}}{(n_{i}-1)!\prod_{k=1}^{p,(i)}(\alpha_{k}-\alpha_{i})_{n_{k}}}\\\
\times\dfrac{\Gamma(\alpha_{i}+\beta+|\vec{n}|)}{\Gamma(\beta+|\vec{n}|)\Gamma(\alpha_{i}+1)}\sum_{l=0}^{n_{i}-1}\dfrac{(-n_{i}+1)_{l}}{l!}\dfrac{(\alpha_{i}+\beta+|\vec{n}|)_{l}}{(\alpha_{i}+1)_{l}}\prod_{k=1}^{p,(i)}\dfrac{(\alpha_{i}-\alpha_{k}-n_{k}+1)_{l}}{(\alpha_{i}-\alpha_{k}+1)_{l}}\\\
\times\underbrace{\int_{0}^{1}x^{\alpha_{i}+j+l}(1-x)^{\beta}\operatorname{d}x}_{\frac{\Gamma(\beta+1)\Gamma(\alpha_{i}+j+l+1)}{\Gamma(\alpha_{i}+\beta+j+l+2)}\,\text{by
\eqref{MomentJP}}}\end{multlined}(-1)^{|\vec{n}|-1}\dfrac{\prod_{k=1}^{p}(\alpha_{k}+\beta+|\vec{n}|)_{n_{k}}}{(n_{i}-1)!\prod_{k=1}^{p,(i)}(\alpha_{k}-\alpha_{i})_{n_{k}}}\\\
\times\dfrac{\Gamma(\alpha_{i}+\beta+|\vec{n}|)}{\Gamma(\beta+|\vec{n}|)\Gamma(\alpha_{i}+1)}\sum_{l=0}^{n_{i}-1}\dfrac{(-n_{i}+1)_{l}}{l!}\dfrac{(\alpha_{i}+\beta+|\vec{n}|)_{l}}{(\alpha_{i}+1)_{l}}\prod_{k=1}^{p,(i)}\dfrac{(\alpha_{i}-\alpha_{k}-n_{k}+1)_{l}}{(\alpha_{i}-\alpha_{k}+1)_{l}}\\\
\times\underbrace{\int_{0}^{1}x^{\alpha_{i}+j+l}(1-x)^{\beta}\operatorname{d}x}_{\frac{\Gamma(\beta+1)\Gamma(\alpha_{i}+j+l+1)}{\Gamma(\alpha_{i}+\beta+j+l+2)}\,\text{by
\eqref{MomentJP}}}\\\
&=\begin{multlined}(-1)^{|\vec{n}|-1}\dfrac{\prod_{k=1}^{p}(\alpha_{k}+\beta+|\vec{n}|)_{n_{k}}}{(n_{i}-1)!\prod_{k=1}^{p,(i)}(\alpha_{k}-\alpha_{i})_{n_{k}}}\dfrac{\Gamma(\alpha_{i}+\beta+|\vec{n}|)(\alpha_{i}+1)_{j}}{\Gamma(\alpha_{i}+\beta+j+2)(\beta+1)_{|\vec{n}|-1}}\\\
\times{}_{p+2}F_{p+1}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-n_{i}+1\mskip
8.0mu\alpha_{i}+j+1\mskip 8.0mu\alpha_{i}+\beta+|\vec{n}|\mskip
8.0mu(\alpha_{i}+1)\vec{e}_{p-1}-\vec{\alpha}^{(i)}-\vec{n}^{(i)}}{\alpha_{i}+1\mskip
8.0mu\alpha_{i}+\beta+j+2\mskip
8.0mu(\alpha_{i}+1)\vec{e}_{p-1}-\vec{\alpha}^{(i)}};1\biggr{]}.\end{multlined}(-1)^{|\vec{n}|-1}\dfrac{\prod_{k=1}^{p}(\alpha_{k}+\beta+|\vec{n}|)_{n_{k}}}{(n_{i}-1)!\prod_{k=1}^{p,(i)}(\alpha_{k}-\alpha_{i})_{n_{k}}}\dfrac{\Gamma(\alpha_{i}+\beta+|\vec{n}|)(\alpha_{i}+1)_{j}}{\Gamma(\alpha_{i}+\beta+j+2)(\beta+1)_{|\vec{n}|-1}}\\\
\times{}_{p+2}F_{p+1}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-n_{i}+1\mskip
8.0mu\alpha_{i}+j+1\mskip 8.0mu\alpha_{i}+\beta+|\vec{n}|\mskip
8.0mu(\alpha_{i}+1)\vec{e}_{p-1}-\vec{\alpha}^{(i)}-\vec{n}^{(i)}}{\alpha_{i}+1\mskip
8.0mu\alpha_{i}+\beta+j+2\mskip
8.0mu(\alpha_{i}+1)\vec{e}_{p-1}-\vec{\alpha}^{(i)}};1\biggr{]}.\end{aligned}$

In this scenario, the orthogonality conditions (2) that we need to establish
are equivalent to the following hypergeometric summation formula.

(9)
$(-1)^{|\vec{n}|-1}\dfrac{\prod_{k=1}^{p}(\alpha_{k}+\beta+|\vec{n}|)_{n_{k}}}{(\beta+1)_{|\vec{n}|-1}}\sum_{i=1}^{p}\dfrac{\Gamma(\alpha_{i}+\beta+|\vec{n}|)(\alpha_{i}+1)_{j}}{(n_{i}-1)!\prod_{k=1}^{p,(i)}(\alpha_{k}-\alpha_{i})_{n_{k}}\Gamma(\alpha_{i}+\beta+j+2)}\\\
\times{}_{p+2}F_{p+1}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-n_{i}+1\mskip
8.0mu\alpha_{i}+j+1\mskip 8.0mu\alpha_{i}+\beta+|\vec{n}|\mskip
8.0mu(\alpha_{i}+1)\vec{e}_{p-1}-\vec{\alpha}^{(i)}-\vec{n}^{(i)}}{\alpha_{i}+1\mskip
8.0mu\alpha_{i}+\beta+j+2\mskip
8.0mu(\alpha_{i}+1)\vec{e}_{p-1}-\vec{\alpha}^{(i)}};1\biggr{]}\\\
=\begin{cases}0,\;\text{if}\;j\in\\{0,\ldots,|\vec{n}|-2\\},\\\
1,\;\text{if}\;j=|\vec{n}|-1.\end{cases}$

We will now apply Equation (5) to the preceding ${}_{p+2}F_{p+1}$
hypergeometric function.

For $j\in\\{0,\ldots,|n|-2\\}$, in accordance with the notation introduced in
Lemma 1, we can identify

$\displaystyle
a=-n_{i}+1;\quad(f_{1},f_{2})=(\alpha_{i}+1,\alpha_{i}+\beta+j+2),\quad(m_{1},m_{2})=(j,|\vec{n}|-2-j);$
$\displaystyle(b_{1},\ldots,b_{p-1})=(\alpha_{i}+1)\vec{e}_{p-1}-\vec{\alpha}^{(i)}-\vec{n}^{(i)},\quad(k_{1},\ldots,k_{p-1})=\vec{n}^{(i)}.$

The condition
$\operatorname{Re}(k_{1}+\cdots+k_{p-1}-a-m_{1}-m_{2})=\operatorname{Re}(|\vec{n}|-n_{i}+n_{i}-1-j-|\vec{n}|+2+j)=1>0$
is met, allowing us to apply Equation (5). After some calculations, we can
express the ${}_{p+2}F_{p+1}$ hypergeometric series in (9) as follows:

$\dfrac{\Gamma(\alpha_{i}+\beta+|\vec{n}|)(\alpha_{i}+1)_{j}}{(n_{i}-1)!\prod_{k=1}^{p,(i)}(\alpha_{k}-\alpha_{i})_{n_{k}}\Gamma(\alpha_{i}+\beta+j+2)}\\\
\times{}_{p+2}F_{p+1}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-n_{i}+1\mskip
8.0mu\alpha_{i}+j+1\mskip 8.0mu\alpha_{i}+\beta+|\vec{n}|\mskip
8.0mu(\alpha_{i}+1)\vec{e}_{p-1}-\vec{\alpha}^{(i)}-\vec{n}^{(i)}}{\alpha_{i}+1\mskip
8.0mu\alpha_{i}+\beta+j+2\mskip
8.0mu(\alpha_{i}+1)\vec{e}_{p-1}-\vec{\alpha}^{(i)}};1\biggr{]}\\\
=-\sum_{q=1,q\neq
i}^{p}\dfrac{\Gamma(\alpha_{q}+\beta+|\vec{n}|)(\alpha_{q}+1)_{j}}{(n_{q}-1)!\prod_{k=1}^{p,(q)}(\alpha_{k}-\alpha_{q})_{n_{k}}\Gamma(\alpha_{q}+\beta+j+2)}\\\
\times{}_{p+2}F_{p+1}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-n_{q}+1\mskip
8.0mu\alpha_{q}+j+1\mskip 8.0mu\alpha_{q}+\beta+|\vec{n}|\mskip
8.0mu(\alpha_{q}+1)\vec{e}_{p-1}-\vec{\alpha}^{(q)}-\vec{n}^{(q)}}{\alpha_{q}+1\mskip
8.0mu\alpha_{q}+\beta+j+2\mskip
8.0mu(\alpha_{q}+1)\vec{e}_{p-1}-\vec{\alpha}^{(q)}};1\biggr{]}.$

The left-hand side of Equation (9) equals 0 for $j=0,\ldots,|\vec{n}|-2$.

Now, when $j=|\vec{n}|-1$, referring to the notation introduced in Lemma 1, we
can identify:

$\displaystyle a=-n_{i}+1;\quad f=\alpha_{i}+1,\quad m=|\vec{n}|-1,$
$\displaystyle(b_{1},\ldots,b_{p})=\left(\alpha_{i}+\beta+|\vec{n}|,\,(\alpha_{i}+1)\vec{e}_{p-1}-\vec{\alpha}^{(i)}-\vec{n}^{(i)}\right),$
$\displaystyle(k_{1},\ldots,k_{p})=\left(1,\,\vec{n}^{(i)}\right).$

The condition
$\operatorname{Re}(k_{1}+\cdots+k_{p}-a-m)=\operatorname{Re}(1+|\vec{n}|-n_{i}+n_{i}-1-|\vec{n}|+1)=1>0$
is met, allowing us to apply Equation (5). This time, we introduce an
additional parameter $b$, which results in an extra term in the decomposition.
After some calculations, we can express the ${}_{p+2}F_{p+1}$ function in (9)
as follows:

$\dfrac{\Gamma(\alpha_{i}+\beta+|\vec{n}|)(\alpha_{i}+1)_{|\vec{n}|-1}}{(n_{i}-1)!\prod_{k=1}^{p,(i)}(\alpha_{k}-\alpha_{i})_{n_{k}}\Gamma(\alpha_{i}+\beta+|\vec{n}|+1)}\\\
\times{}_{p+2}F_{p+1}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-n_{i}+1\mskip
8.0mu\alpha_{i}+|\vec{n}|\mskip 8.0mu\alpha_{i}+\beta+|\vec{n}|\mskip
8.0mu(\alpha_{i}+1)\vec{e}_{p-1}-\vec{\alpha}^{(i)}-\vec{n}^{(i)}}{\alpha_{i}+1\mskip
8.0mu\alpha_{i}+\beta+|\vec{n}|+1\mskip
8.0mu(\alpha_{i}+1)\vec{e}_{p-1}-\vec{\alpha}^{(i)}};1\biggr{]}\\\
=(-1)^{|\vec{n}|-1}\dfrac{(\beta+1)_{|\vec{n}|-1}}{\prod_{k=1}^{p}(\alpha_{k}+\beta+|\vec{n}|)_{n_{k}}}-\sum_{q=1,q\neq
i}^{p}\dfrac{\Gamma(\alpha_{q}+\beta+|\vec{n}|)(\alpha_{q}+1)_{|\vec{n}|-1}}{(n_{q}-1)!\prod_{k=1}^{p,(q)}(\alpha_{k}-\alpha_{q})_{n_{k}}\Gamma(\alpha_{q}+\beta+|\vec{n}|+1)}\\\
\times{}_{p+2}F_{p+1}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-n_{q}+1\mskip
8.0mu\alpha_{q}+|\vec{n}|\mskip 8.0mu\alpha_{q}+\beta+|\vec{n}|\mskip
8.0mu(\alpha_{q}+1)\vec{e}_{p-1}-\vec{\alpha}^{(q)}-\vec{n}^{(q)}}{\alpha_{q}+1\mskip
8.0mu\alpha_{q}+\beta+|\vec{n}|+1\mskip
8.0mu(\alpha_{q}+1)\vec{e}_{p-1}-\vec{\alpha}^{(q)}};1\biggr{]}.$

For $j=|\vec{n}|-1$, the left-hand side of Equation (9) equals $1$. ∎

##  3\. Hypergeometric expressions for type I Laguerre of first kind multiple
orthogonal polynomials

In this case, we define the weight functions as:

(10) $\displaystyle w_{i}(x;\alpha_{i})=e^{-x}x^{\alpha_{i}};$ $\displaystyle
i=1,\ldots,p;$ $\displaystyle\operatorname{d}\mu(x)=\operatorname{d}x;$
$\displaystyle\Delta=[0,\infty).$

Here $\vec{\alpha}=(\alpha_{1},\ldots,\alpha_{p})$ with
$\alpha_{1},\ldots,\alpha_{p}>-1$, and, to establish an AT system, we require
that $\alpha_{i}-\alpha_{j}\not\in\mathbb{Z}$ for $i\neq j$.

In [6, §7], it was demonstrated that the type I polynomials for $p=2$ could be
derived from the Jacobi–Piñeiro polynomials through the following limit:

$L^{(i)}_{(n_{1},n_{2})}(x;\alpha_{1},\alpha_{2})\\\
=\lim_{\beta\rightarrow\infty}\dfrac{\Gamma(\beta+n_{1}+n_{2})}{(\alpha_{1}+\beta+n_{1}+n_{2})_{n_{1}}(\alpha_{2}+\beta+n_{1}+n_{2})_{n_{2}}\Gamma(\alpha_{i}+\beta+n_{1}+n_{2})}P_{(n_{1},n_{2})}^{(i)}\left(\dfrac{x}{\beta};\alpha_{1},\alpha_{2},\beta\right).$

This yields the expression:

$L_{(n_{1},n_{2})}^{(i)}(x;\alpha_{1},\alpha_{2})\\\
=(-1)^{n_{1}+n_{2}-1}\dfrac{1}{(n_{i}-1)!\Gamma(\alpha_{i}+1)(\hat{\alpha}_{i}-\alpha_{i})_{\hat{n}_{i}}}{}_{2}F_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-n_{i}+1\mskip
8.0mu\alpha_{i}-\hat{\alpha}_{i}-\hat{n}_{i}+1}{\alpha_{i}+1\mskip
8.0mu\alpha_{i}-\hat{\alpha}_{i}+1};x\biggr{]}.$

Here, $\hat{\alpha}_{i}=\delta_{i,2}\alpha_{1}+\delta_{i,1}\alpha_{2}$, and
$\hat{n}_{i}=\delta_{i,2}n_{1}+\delta_{i,1}n_{2}$ with $i\in\\{1,2\\}$.

Once again, in an attempt to generalize this expression, we arrive at:

######  Theorem 3.

The Laguerre of the first kind polynomials of type I for $p\geq 2$ are
expressed as:

(11)
$L^{(i)}_{\vec{n}}(x;\alpha_{1},\ldots,\alpha_{p})=(-1)^{|\vec{n}|-1}\dfrac{1}{(n_{i}-1)!\prod_{i\neq
k=1}^{p}(\alpha_{k}-\alpha_{i})_{n_{k}}\Gamma(\alpha_{i}+1)}\\\
\times{}_{p}F_{p}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-n_{i}+1\mskip
8.0mu(\alpha_{i}+1)\vec{e}_{p-1}-\vec{\alpha}^{(i)}-\vec{n}^{(i)}}{\alpha_{i}+1\mskip
8.0mu(\alpha_{i}+1)\vec{e}_{p-1}-\vec{\alpha}^{(i)}};x\biggr{]}.$

These polynomials are derived by applying the limit to the Jacobi–Piñeiro type
I polynomials (8), as described in:

(12) $L^{(i)}_{\vec{n}}(x;\alpha_{1},\ldots,\alpha_{p})\\\
=\lim_{\beta\rightarrow\infty}\dfrac{\Gamma(\beta+|\vec{n}|)}{\prod_{k=1}^{p}(\alpha_{k}+\beta+|\vec{n}|)_{n_{k}}\Gamma(\alpha_{i}+\beta+|\vec{n}|)}P_{\vec{n}}^{(i)}\left(\dfrac{x}{\beta};\alpha_{1},\ldots,\alpha_{p},\beta\right).$

###### Proof.

In this context, we will leverage the fact that the polynomials defined in
(11) can be obtained through Jacobi–Piñeiro as shown in (8) via the limit
(12). In Theorem 2, we have established the following orthogonality conditions
for the Jacobi–Piñeiro polynomials $P_{\vec{n}}^{(i)}$:

$\displaystyle\sum_{i=1}^{p}\int_{0}^{1}x^{j}P^{(i)}_{\vec{n}}(x;\alpha_{1},\ldots,\alpha_{p},\beta)x^{\alpha_{i}}(1-x)^{\beta}\operatorname{d}x=\begin{cases}0,\;\text{if}\;j\in\\{0,\ldots,|\vec{n}|-2\\},\\\
1,\;\text{if}\;j=|\vec{n}|-1.\end{cases}$

This expression can be reformulated by substituting the variable $x$ with
$\frac{x}{\beta}$ and introducing certain constants as follows:

$\sum_{i=1}^{p}\dfrac{1}{\beta^{|\vec{n}|-1-j}}\dfrac{\prod_{k=1}^{p}(\alpha_{k}+\beta+|\vec{n}|)_{n_{k}}\Gamma(\alpha_{i}+\beta+|\vec{n}|)}{\Gamma(\beta+|\vec{n}|)\beta^{\alpha_{i}+j+1}}\\\
\times\int_{0}^{\beta}x^{j}\dfrac{\Gamma(\beta+|\vec{n}|)}{\prod_{k=1}^{p}(\alpha_{k}+\beta+|\vec{n}|)_{n_{k}}\Gamma(\alpha_{i}+\beta+|\vec{n}|)}P^{(i)}_{\vec{n}}\left(\dfrac{x}{\beta};\alpha_{1},\ldots,\alpha_{p},\beta\right)x^{\alpha_{i}}\left(1-\dfrac{x}{\beta}\right)^{\beta}\operatorname{d}x\\\
=\begin{cases}{0},\;\text{if}\;j\in\\{0,\ldots,|\vec{n}|-2\\},\\\
{1},\;\text{if}\;j=|\vec{n}|-1.\\\ \end{cases}$

Now, our objective is to apply the limit as $\beta\rightarrow\infty$ to both
sides of the preceding equation. To ensure that the limit can be interchanged
with the integral, we need to establish that the modulus of the expression
within the integral is bounded for all $\beta>0$.

On one hand, we can establish that:

$\displaystyle\left(1-\dfrac{x}{\beta}\right)^{\beta}I_{[0,\beta)}\leq\mathrm{e}^{-x},$
$\displaystyle\beta>0,$ $\displaystyle x\in\mathbb{R}^{+}.$

On the other hand, based on its definition, the coefficients of the
polynomial:

$\displaystyle\dfrac{\Gamma(\beta+|\vec{n}|)}{\prod_{k=1}^{p}(\alpha_{k}+\beta+|\vec{n}|)_{n_{k}}\Gamma(\alpha_{i}+\beta+|\vec{n}|)}P^{(i)}_{\vec{n}}\left(\dfrac{x}{\beta};\alpha_{1},\ldots,\alpha_{p},\beta\right)$

are bounded by the coefficients of the respective polynomial
$L_{\vec{n}}^{(i)}(x;\alpha_{1},\ldots,\alpha_{p})$, cf. (11), for all
$\beta>0$ and $x\in\mathbb{R}^{+}$. This implies that the modulus of the
aforementioned polynomial can be bounded for all $\beta>0$ and
$x\in\mathbb{R}^{+}$ by the sum of the moduli of the coefficients of
$L_{\vec{n}}^{(i)}(x;\alpha_{1},\ldots,\alpha_{p})$ plus an additive constant.
This entire sum can then be multiplied by $x^{n_{i}-1}$ if $x>1$ or by $x^{0}$
if $x\leq 1$.

So, we have shown that the expression within the integral is bounded for all
$\beta>0$ and all $x\in\mathbb{R}^{+}$. By applying Lebesgue’s dominated
convergence theorem, we can interchange the limit as $\beta\rightarrow\infty$

$\sum_{i=1}^{p}\overbrace{\lim_{\beta\rightarrow\infty}\dfrac{1}{\beta^{|\vec{n}|-1-j}}\dfrac{\prod_{k=1}^{p}(\alpha_{k}+\beta+|\vec{n}|)_{n_{k}}\Gamma(\alpha_{i}+\beta+|\vec{n}|)}{\Gamma(\beta+|\vec{n}|)\beta^{\alpha_{i}+j+1}}}^{1}\\\
\times\int_{0}^{\infty}x^{j}\underbrace{\lim_{\beta\rightarrow\infty}\dfrac{\Gamma(\beta+|\vec{n}|)}{\prod_{k=1}^{p}(\alpha_{k}+\beta+|\vec{n}|)_{n_{k}}\Gamma(\alpha_{i}+\beta+|\vec{n}|)}P^{(i)}_{\vec{n}}\left(\dfrac{x}{\beta};\alpha_{1},\ldots,\alpha_{p},\beta\right)}_{L^{(i)}_{\vec{n}}(x;\alpha_{1},\ldots,\alpha_{p})\,\text{by
\eqref{LaguerreITypeIasLimitJPTypeI}}}x^{\alpha_{i}}\\\
\times\underbrace{\lim_{\beta\rightarrow\infty}\left(1-\dfrac{x}{\beta}\right)^{\beta}}_{\mathrm{e}^{-x}}\operatorname{d}x=\begin{cases}0,\;\text{if}\;j\in\\{0,\ldots,|\vec{n}|-2\\},\\\
1,\;\text{if}\;j=|\vec{n}|-1.\end{cases}$

We have established the orthogonality conditions (2) for the polynomials (11)
with respect to the weight functions (10). ∎

##  4\. Conclusions and outlook

In this paper, we have successfully derived explicit generalized
hypergeometric expressions for two families of type I multiple orthogonal
polynomials with an arbitrary number of weights. This finding is particularly
pertinent in contexts where type I polynomials play a crucial role, such as
their application in Markov chains with transition matrices featuring $p$
subdiagonals.

For future research, an essential objective is to identify similar
hypergeometric expressions using Kampé de Fériet functions for Hahn multiple
orthogonal polynomials with $p$ measures. This exploration within the Askey
scheme will enable us to acquire corresponding hypergeometric expressions for
the multiple orthogonal polynomial of type I families under Hahn.

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