On a Family of Multivariate Modified Humbert Polynomials

Abstract

This paper attempts to present a multivariable extension of generalized Humbert polynomials. The results obtained here include various families of multilinear and multilateral generating functions, miscellaneous properties, and also some special cases for these multivariable polynomials.

1. Introduction

The generalized Humbert polynomials are generated by [1]

$$\begin{array}{c}{\left(C-mxt+y{t}^m\right)}^p=\sum _{n=\mathrm{0}}^{\infty }{P}_n\left(m,x,y,p,C\right)t^n,\end{array}$$ (1)

where m is a positive integer and other parameters are unrestricted in general (see also [2, pages 77, 86] and [3, 4]). This definition includes many well-known special polynomials such as Humbert, Louville, Gegenbauer, Legendre, Tchebycheff, Pincherle, and Kinney polynomials.

In this paper, we consider the following multivariable extension of the generalized Humbert polynomials which are completely different from the polynomials introduced in [5]. This class of polynomials is generated by

$$\begin{array}{c}{\left(\mathrm{1}-\sum _{i=\mathrm{1}}^r\left(m_i{x}_i{t}_i-y_i{t}_i^{m_i}\right)\right)}^{-\alpha }\\ \hspace{1em}\hspace{1em}=\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}(\mathbf{x},\mathbf{y},\mathbf{m})t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r},\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(\left|\sum _{i=\mathrm{1}}^r\left(m_i{x}_i{\text{t}}_i-y_i{t}_i^{m_i}\right)\right|<\mathrm{1}\right),\end{array}$$ (2)

where x = (x ₁,…, x _r),  y = (y ₁,…, y _r), m = (m ₁,…, m _r) and m _i  (i = 1,2,…, r) is a positive integer. It follows from (2) that

$$\begin{array}{c}{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \hspace{1em}=\sum _{k_{\mathrm{1}}=\mathrm{0}}^{\left[n_{\mathrm{1}}/m_{\mathrm{1}}\right]}\cdots \sum _{k_r=\mathrm{0}}^{\left[n_r/m_r\right]}\frac{{\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r-\left(m_{\mathrm{1}}-\mathrm{1}\right)k_{\mathrm{1}}-\cdots -\left(m_r-\mathrm{1}\right)k_r}}{k_{\mathrm{1}}!\cdots k_r!\left(n_{\mathrm{1}}-m_{\mathrm{1}}{k}_{\mathrm{1}}\right)!\cdots \left(n_r-m_r{k}_r\right)!}\\ \hspace{1em}\hspace{1em}\times {\left(-\mathrm{1}\right)}^{k_{\mathrm{1}}+\cdots +k_r}{\left(m_{\mathrm{1}}{x}_{\mathrm{1}}\right)}^{n_{\mathrm{1}}-m_{\mathrm{1}}{k}_{\mathrm{1}}}\cdots {\left(m_r{x}_r\right)}^{n_r-m_r{k}_r}{y}_{\mathrm{1}}^{k_{\mathrm{1}}}\cdots y_r^{k_r},\end{array}$$ (3)

where (λ)_k : = λ(λ + 1) ⋯ (λ + k − 1), (λ)₀ : = 1 is the Pochhammer symbol.

The aim of this paper is to derive various families of multilinear and multilateral generating functions and to give several recurrence relations and expansions in the series of orthogonal polynomials for the family of multivariable polynomials given explicitly by (3). We present some special cases of our results and also obtain some other properties for these special cases.

2. Bilinear and Bilateral Generating Functions

In this section, with the help of the similar method as considered in [5–9], we derive several families of bilinear and bilateral generating functions for the family of multivariable polynomials generated by (2) and given explicitly by (3).

We begin by stating the following theorem.

Theorem 1 —

Corresponding to an identically nonvanishing function Ω _μ(z) of s complex variables z ₁,…, z _s  (s ∈ ℕ) and of complex order μ, let

$$\begin{array}{c}{\mathrm{\Lambda }}_{\mu ,\nu }\left(\mathbf{z};w\right):=\sum _{k=\mathrm{0}}^{\infty }{a}_k{\Omega }_{\mu +\nu k}\left(\mathbf{z}\right)w^k,\end{array}$$ (4)

where (a _k ≠ 0,   μ, ν ∈ ℂ); z = (z ₁,…, z _s). Then, for p ∈ ℕ; x = (x ₁,…, x _r); y = (y ₁,…, y _r); m = (m ₁,…, m _r), m _i ∈ ℕ (i = 1,2,…, r), one has

$$\begin{array}{c}\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }\sum _{k=\mathrm{0}}^{\left[n_{\mathrm{1}}/p\right]\mathrm{\hspace{0.17em}\hspace{0.17em}}}{a}_k{\mathrm{\Theta }}_{n_{\mathrm{1}}-pk,n_{\mathrm{2}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times t_{\mathrm{2}}^{n_{\mathrm{2}}}\cdots t_r^{n_r}{\Omega }_{\mu +\nu k}\left(\mathbf{z}\right){\eta }^k{t}_{\mathrm{1}}^{n_{\mathrm{1}}-pk}\\ \hspace{1em}\hspace{1em}={\left(\mathrm{1}-\sum _{i=\mathrm{1}}^r\left(m_i{x}_i{t}_i-y_i{t}_i^{m_i}\right)\right)}^{-\alpha }{\mathrm{\Lambda }}_{\mu ,\nu }\left(\mathbf{z};\eta \right)\end{array}$$ (5)

provided that each member of (5) exists.

Proof —

For convenience, let S denote the first member of the assertion (5) of Theorem 1. Replacing n ₁ by n ₁ + pk, we may write that

$$\begin{array}{c}S=\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }\hspace{0.17em}‍\sum _{k=\mathrm{0}}^{\infty }{a}_k\mathrm{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\Theta }}_{n_{\mathrm{1}},n_{\mathrm{2}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right){\Omega }_{\mu +\nu k}\left(\mathbf{z}\right){\eta }^k{t}_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}\\ =\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{\mathrm{\Theta }}_{n_{\mathrm{1}},n_{\mathrm{2}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}\sum _{k=\mathrm{0}}^{\infty }{a}_k{\Omega }_{\mu +\nu k}\left(\mathbf{z}\right){\eta }^k\\ ={\left(\mathrm{1}-\sum _{i=\mathrm{1}}^r\left(m_i{x}_i{t}_i-y_i{t}_i^{m_i}\right)\right)}^{-\alpha }{\mathrm{\Lambda }}_{\mu ,\nu }\left(\mathbf{z};\eta \right),\end{array}$$ (6)

which completes the proof.

By using a similar idea, we also get the next result immediately.

Theorem 2 —

For a nonvanishing function φ _λ1,…,λr(z) of s complex variables z ₁,…, z _s  (s ∈ ℕ) and for p _i ∈ ℕ, λ _i, η _i ∈ ℂ, z = (z ₁,…, z _s), w = (w ₁,…, w _r), λ : = (λ ₁,…, λ _r),  η : = (η ₁,…, η _r) let

$$\begin{array}{c}{\Omega }_{\lambda ,\eta ,\alpha ,\beta ,\mathbf{m}}^{\mathbf{n},\mathbf{p}}(\mathbf{x},\mathbf{y};\mathbf{z};\mathbf{w})\\ \hspace{1em}:=\sum _{k_{\mathrm{1}}=\mathrm{0}}^{\left[n_{\mathrm{1}}/p_{\mathrm{1}}\right]}\cdots \sum _{k_r=\mathrm{0}}^{\left[n_r/p_r\right]}{a}_{k_{\mathrm{1}},\dots ,k_r}{\mathrm{\Theta }}_{n_{\mathrm{1}}-p_{\mathrm{1}}{k}_{\mathrm{1}},\dots ,n_r-p_r{k}_r}^{\left(\alpha +\beta \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\phi }_{{\lambda }_{\mathrm{1}}+{\eta }_{\mathrm{1}}{k}_{\mathrm{1}},\dots ,{\lambda }_r+{\eta }_r{k}_r}\left(\mathbf{z}\right)w_{\mathrm{1}}^{k_{\mathrm{1}}}\cdots w_r^{k_r},\end{array}$$ (7)

where a _k1,…,kr ≠ 0; n _i, (i = 1,2,…, r) ∈ ℕ ₀; ℕ ₀ : = ℕ ∪ {0}. Then, one has

$$\begin{array}{c}\sum _{l_{\mathrm{1}}=\mathrm{0}}^{n_{\mathrm{1}}}\cdots \sum _{l_r=\mathrm{0}}^{n_r}\sum _{k_{\mathrm{1}}=\mathrm{0}}^{\left[l_{\mathrm{1}}/p_{\mathrm{1}}\right]}\cdots \sum _{k_r=\mathrm{0}}^{\left[l_r/p_r\right]}{a}_{k_{\mathrm{1}},\dots ,k_r}{\mathrm{\Theta }}_{n_{\mathrm{1}}-l_{\mathrm{1}},\dots ,n_r-l_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \times {\mathrm{\Theta }}_{l_{\mathrm{1}}-p_{\mathrm{1}}{k}_{\mathrm{1}},\dots ,l_r-p_r{k}_r}^{\left(\beta \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \times {\phi }_{{\lambda }_{\mathrm{1}}+{\eta }_{\mathrm{1}}{k}_{\mathrm{1}},\dots ,{\lambda }_r+{\eta }_r{k}_r}\left(\mathbf{z}\right)\\ \times w_{\mathrm{1}}^{k_{\mathrm{1}}}\cdots w_r^{k_r}\\ \hspace{1em}={\Omega }_{\lambda ,\eta ,\alpha ,\beta ,\mathbf{m}}^{\mathbf{n},\mathbf{p}}\left(\mathbf{x},\mathbf{y};\mathbf{z};\mathbf{w}\right)\end{array}$$ (8)

provided that each member of (8) exists.

3. Special Cases

As an application of the above theorems, when the multivariable function Ω _μ+νk(z),  z = (z ₁,…, z _s),  k ∈ ℕ ₀, s ∈ ℕ, is expressed in terms of simpler functions of one and more variables, then we can give further applications of the above theorems. We first set

$$\begin{array}{c}s=r,\hspace{1em}\hspace{1em}{\Omega }_{\mu +\nu k}\left(\mathbf{z}\right)={\text{g}}_{\mu +\nu k}^{\left({\beta }_{\mathrm{1}},\dots ,{\beta }_r\right)}\left(\mathbf{z}\right)\end{array}$$ (9)

in Theorem 1, where the Chan-Chyan-Srivastava polynomials g_μ+νk ^{(β₁,…,β_r)}(z) [10] are generated by

$$\begin{array}{c}\prod _{i=\mathrm{1}}^r{\left(\mathrm{1}-x_{i}t\right)}^{-{\alpha }_i}=\sum _{n=\mathrm{0}}^{\infty }{\text{g}}_n^{\left({\alpha }_{\mathrm{1}},\dots ,{\alpha }_r\right)}\left(x_{\mathrm{1}},\dots ,x_r\right)t^n\\ \left({\alpha }_i\in ℂ\left(i=\mathrm{1,2},\dots ,r\right);\left|t\right|<\min \left\{{\left|x_{\mathrm{1}}\right|}^{-\mathrm{1}},\dots ,{\left|x_r\right|}^{-\mathrm{1}}\right\}\right).\end{array}$$ (10)

We are thus led to the following result which provides a class of bilateral generating functions for the Chan-Chyan-Srivastava polynomials and the family of multivariable polynomials given explicitly by (3).

Corollary 3 —

If Λ_μ,ν(z; w): = ∑_k=0 ^∞ a _k  g  _μ+νk ^{(β₁,…,β_r)}(z)w ^k, a _k ≠ 0, μ, ν ∈ ℂ, z = (z ₁,…, z _r) then, one has

$$\begin{array}{c}\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }\sum _{k=\mathrm{0}}^{\left[n_{\mathrm{1}}/p\right]}{a}_k{\mathrm{\Theta }}_{n_{\mathrm{1}}-pk,n_{\mathrm{2}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \times t_{\mathrm{2}}^{n_{\mathrm{2}}}\cdots t_r^{n_r}\hspace{0.17em}{\text{g}\hspace{0.17em}}_{\mu +\nu k}^{\left({\beta }_{\mathrm{1}},\dots ,{\beta }_r\right)}\left(\mathbf{z}\right){\eta }^k{t}_{\mathrm{1}}^{n_{\mathrm{1}}-pk}\\ ={\left(\mathrm{1}-\sum _{i=\mathrm{1}}^r\left(m_i{x}_i{t}_i-y_i{t}_i^{m_i}\right)\right)}^{-\alpha }{\mathrm{\Lambda }}_{\mu ,\nu }\left(\mathbf{z};\eta \right)\end{array}$$ (11)

provided that each member of (11) exists.

Remark 4 —

Using the generating relation (10) for the Chan-Chyan-Srivastava polynomials and getting a _k = 1,  μ = 0, ν = 1, we find that

$$\begin{array}{c}\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }\hspace{0.17em}\sum _{k=\mathrm{0}}^{\left[n_{\mathrm{1}}/p\right]}{\mathrm{\Theta }}_{n_{\mathrm{1}}-pk,n_{\mathrm{2}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)t_{\mathrm{2}}^{n_{\mathrm{2}}}\cdots t_r^{n_r}{\text{g}}_k^{\left({\beta }_{\mathrm{1}},\dots ,{\beta }_r\right)}\left(\mathbf{z}\right)\\ \times {\eta }^k{t}_{\mathrm{1}}^{n_{\mathrm{1}}-pk}\\ \hspace{1em}\hspace{1em}={\left(\mathrm{1}-\sum _{i=\mathrm{1}}^r\left(m_i{x}_i{t}_i-y_i{t}_i^{m_i}\right)\right)}^{-\alpha }\prod _{i=\mathrm{1}}^r{\left(\mathrm{1}-z_i\eta \right)}^{-{\beta }_i},\\ \left|\eta \right|<\min \left\{{\left|z_{\mathrm{1}}\right|}^{-\mathrm{1}},\dots ,{\left|z_r\right|}^{-\mathrm{1}}\right\},\left|\sum _{i=\mathrm{1}}^r\left(m_i{x}_i{t}_i-y_i{t}_i^{m_i}\right)\right|<\mathrm{1}.\end{array}$$ (12)

On the other hand, choosing s = 2r  and

$$\begin{array}{c}{\phi }_{{\lambda }_{\mathrm{1}}+{\eta }_{\mathrm{1}}{k}_{\mathrm{1}},\dots ,{\lambda }_r+{\eta }_r{k}_r}\left(\mathbf{z}\right)={\mathrm{\Theta }}_{{\lambda }_{\mathrm{1}}+{\eta }_{\mathrm{1}}{k}_{\mathrm{1}},\dots ,{\lambda }_r+{\eta }_r{k}_r}^{\left(\gamma \right)}\left(\mathbf{u},\mathbf{v},\mathbf{m}\right),\end{array}$$ (13)

λ _i, η _i  (i = 1,2,…, r) ∈ ℕ ₀,  u = (u ₁,…, u _r), v = (v ₁,…, v _r) in Theorem 2, we obtain the following class of bilinear generating functions for the family of multivariable polynomials given explicitly by (3).

Corollary 5 —

If

$$\begin{array}{c}{\mathrm{\Lambda }}_{\lambda ,\eta ,\alpha ,\beta ,\gamma ,\mathbf{m}}^{\mathbf{n},\mathbf{p}}\left(\mathbf{x},\mathbf{y};\mathbf{u},\mathbf{v};\mathbf{w}\right)\\ \hspace{1em}:=\sum _{k_{\mathrm{1}}=\mathrm{0}}^{\left[n_{\mathrm{1}}/p_{\mathrm{1}}\right]}\cdots \sum _{k_r=\mathrm{0}}^{\left[n_r/p_r\right]}{a}_{k_{\mathrm{1}},\dots ,k_r}{\mathrm{\Theta }}_{n_{\mathrm{1}}-p_{\mathrm{1}}{k}_{\mathrm{1}},\dots ,n_r-p_r{k}_r}^{\left(\alpha +\beta \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\mathrm{\hspace{0.17em}\hspace{0.17em}}\\ \times {\mathrm{\Theta }}_{{\lambda }_{\mathrm{1}}+{\eta }_{\mathrm{1}}{k}_{\mathrm{1}},\dots ,{\lambda }_r+{\eta }_r{k}_r}^{\left(\gamma \right)}\left(\mathbf{u},\mathbf{v},\mathbf{m}\right)w_{\mathrm{1}}^{k_{\mathrm{1}}}\cdots w_r^{k_r},\end{array}$$ (14)

where a _k1,…,kr ≠ 0; p _i ∈ ℕ; n _i, λ _i, η _i ∈ ℕ ₀, i = 1,2,…, r, then, we get

$$\begin{array}{c}\sum _{l_{\mathrm{1}}=\mathrm{0}}^{n_{\mathrm{1}}}\cdots \sum _{l_r=\mathrm{0}}^{n_r}\hspace{0.17em}‍\hspace{0.17em}\sum _{k_{\mathrm{1}}=\mathrm{0}}^{\left[l_{\mathrm{1}}/p_{\mathrm{1}}\right]}\cdots \sum _{k_r=\mathrm{0}}^{\left[l_r/p_r\right]}{a}_{k_{\mathrm{1}},\dots ,k_r}{\mathrm{\Theta }}_{n_{\mathrm{1}}-l_{\mathrm{1}},\dots ,n_r-l_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \times {\mathrm{\Theta }}_{l_{\mathrm{1}}-p_{\mathrm{1}}{k}_{\mathrm{1}},\dots ,l_r-p_r{k}_r}^{\left(\beta \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ {\times \mathrm{\Theta }}_{{\lambda }_{\mathrm{1}}+{\eta }_{\mathrm{1}}{k}_{\mathrm{1}},\dots ,{\lambda }_r+{\eta }_r{k}_r}^{\left(\gamma \right)}\left(\mathbf{u},\mathbf{v},\mathbf{m}\right)\\ \times w_{\mathrm{1}}^{k_{\mathrm{1}}}\cdots w_r^{k_r}\\ \hspace{1em}\hspace{1em}={\mathrm{\Lambda }}_{\lambda ,\eta ,\alpha ,\beta ,\gamma ,\mathbf{m}}^{\mathbf{n},\mathbf{p}}\left(\mathbf{x},\mathbf{y};\mathbf{u},\mathbf{v};\mathbf{w}\right)\end{array}$$ (15)

provided that each member of (15) exists.

Furthermore, for every suitable choice of the coefficients a _k  (k ∈ ℕ ₀), if the multivariable functions Ω _μ+νk(y) and φ _{λ1+η1k1,…,λr+ηrkr}(y),  y = (y ₁,…, y _s), (s ∈ ℕ), are expressed as an appropriate product of several simpler functions, the assertions of Theorems 1 and 2 can be applied in order to derive various families of multilinear and multilateral generating functions for the family of multivariable polynomials given explicitly by (3).

4. Some Miscellaneous Properties

In this section, we now discuss some further properties of the family of multivariable polynomials given by (3). We start with the following theorems.

Theorem 6 —

Let Ψ_n1,…,nr(x, y, m) be a family of functions generated by

$$\begin{array}{c}G\left(m_{\mathrm{1}}{x}_{\mathrm{1}}{t}_{\mathrm{1}}-y_{\mathrm{1}}{t}_{\mathrm{1}}^{m_{\mathrm{1}}},\dots ,m_r{x}_r{t}_r-y_r{t}_r^{m_r}\right)\\ \hspace{1em}\hspace{1em}=\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}.\end{array}$$ (16)

The following relations hold:

$$\begin{array}{c}{n}_i{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)=x_i\frac{\partial }{\partial x_i}{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \hspace{1em}-y_i\frac{\partial }{\partial x_i}{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\end{array}$$ (17)

for n _i ≥ m _i − 1,  i = 1,2,…, r; n _j ≥ 0,  j ≠ i and

$$\begin{array}{c}{n}_i{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)=x_i\frac{\partial }{\partial x_i}{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\end{array}$$ (18)

for n _i = 0,1,…, m _i − 2, i = 1,2,…, r; n _j ≥ 0, j ≠ i. Also, one finds that

$$\begin{array}{c}{m}_i{x}_i\frac{\partial }{\partial y_i}{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)-m_i{y}_i\frac{\partial }{\partial y_i}\\ \times {\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ =-\left(n_i-m_i+\mathrm{1}\right){\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\end{array}$$ (19)

for n _i ≥ m _i − 1, i = 1,2,…, r; n _j ≥ 0, j ≠ i and

$$\begin{array}{c}\frac{\partial }{\partial y_i}{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_r}(\mathbf{x},\mathbf{y},\mathbf{m})=\mathrm{0}\end{array}$$ (20)

for n _i = 0,1,…, m _i − 2,  i = 1,2,…, r; n _j ≥ 0, j ≠ i.

Proof —

Fix i = 1,2,…, r. Then, by differentiating (16) with respect to x _i and t _i, after making necessary calculations we obtain that

$$\begin{array}{c}\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{n}_i{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}\\ \hspace{1em}=\left(x_i-y_i{t}_i^{m_i-\mathrm{1}}\right)\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }\frac{\partial }{\partial x_i}{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}\\ \hspace{1em}=x_i\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }\frac{\partial }{\partial x_i}{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}\\ \hspace{1em}\hspace{1em}-y_i\sum _{\begin{array}{c}{n}_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_{i+\mathrm{1}},\dots ,n_r=\mathrm{0}\\ \left(n_i\ge m_i-\mathrm{1}\right)\end{array}}^{\infty }\frac{\partial }{\partial x_i}{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \times t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}.\end{array}$$ (21)

Comparing the coefficients of t ₁ ^n₁ ⋯ t _r ^n_r, we obtain (17) and (18) for the fixed i. Similarly, if we differentiate (16) with respect to y _i and t _i, we can find the relation (19).

Theorem 7 —

If Ψ_n1,…,nr(x, y, m) is a family of functions generated by (16), then it satisfies the relations

$$\begin{array}{c}-m_i\frac{\partial }{\partial y_i}{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \hspace{1em}\hspace{1em}=\frac{\partial }{\partial x_i}{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i,n_{i+\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\end{array}$$ (22)

for n _i ≥ m _i, i = 1,2,…, r; n _j ≥ 0, j ≠ i and

$$\begin{array}{c}\frac{\partial }{\partial y_i}{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)=\mathrm{0}\end{array}$$ (23)

for n _i = 1,2,…, m _i − 1, i = 1,2,…, r; n _j ≥ 0, j ≠ i.

Proof —

By comparing the derivatives of (16) with respect to x _i and y _i, we have

$$\begin{array}{c}-m_i{t}_i\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }\frac{\partial }{\partial y_i}{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}\\ \hspace{1em}\hspace{1em}=t_i^{m_i}\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }\frac{\partial }{\partial x_i}{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r},\end{array}$$ (24)

which implies that

$$\begin{array}{c}-m_i\frac{\partial }{\partial y_i}{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \hspace{1em}\hspace{1em}=\frac{\partial }{\partial x_i}{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i,n_{i+\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\end{array}$$ (25)

for n _i ≥ m _i, i = 1,2,…, r; n _j ≥ 0, j ≠ i and

$$\begin{array}{c}\frac{\partial }{\partial y_i}{\mathrm{\Psi }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)=\mathrm{0}\end{array}$$ (26)

for n _i = 1,2,…, m _i − 1,  i = 1,2,…, r; n _j ≥ 0, j ≠ i. Thus, the proof is completed.

As a result of these theorems, if we choose

$$\begin{array}{c}G\left(m_{\mathrm{1}}{x}_{\mathrm{1}}{t}_{\mathrm{1}}-y_{\mathrm{1}}{t}_{\mathrm{1}}^{m_{\mathrm{1}}},\dots ,m_r{x}_r{t}_r-y_r{t}_r^{m_r}\right)\\ \hspace{1em}\hspace{1em}={\left(\mathrm{1}-\sum _{i=\mathrm{1}}^r\left(m_i{x}_i{t}_i-y_i{t}_i^{m_i}\right)\right)}^{-\alpha },\end{array}$$ (27)

then we can give some recurrence relations for the family of multivariable polynomials given explicitly by (3). In the view of Theorem 6, we get

Corollary 8 —

For the family of multivariable polynomials generated by (2), the following relations hold

$$\begin{array}{c}{n}_i{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)=x_i\frac{\partial }{\partial x_i}{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ -y_i\frac{\partial }{\partial x_i}{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right),\end{array}$$ (28)

$$\begin{array}{c}{m}_i{x}_i\frac{\partial }{\partial y_i}{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)-m_i{y}_i\frac{\partial }{\partial y_i}\\ \times {\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ =-\left(n_i-m_i+\mathrm{1}\right){\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\end{array}$$ (29)

for n _i ≥ m _i − 1,  i = 1,2,…, r; n _j ≥ 0, j ≠ i. Also, for n _i = 0,1,…, m _i − 2, i = 1,2,…, r; n _j ≥ 0, j ≠ i, we have

$$\begin{array}{c}{n}_i{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)=x_i\frac{\partial }{\partial x_i}{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right),\end{array}$$ (30)

$$\begin{array}{c}\frac{\partial }{\partial y_i}{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)=\mathrm{0}.\end{array}$$ (31)

Corollary 9 —

By summing the relations given by (28) and (29), respectively, for i = 1,2,…, r, we get

$$\begin{array}{c}\sum _{i=\mathrm{1}}^r{n}_i{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \hspace{1em}\hspace{1em}=\sum _{i=\mathrm{1}}^r{x}_i\frac{\partial }{\partial x_i}{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}-\sum _{i=\mathrm{1}}^r{y}_i\frac{\partial }{\partial x_i}{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right),\\ \sum _{i=\mathrm{1}}^r{m}_i{x}_i\frac{\partial }{\partial y_i}{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}-\sum _{i=\mathrm{1}}^r{m}_i{y}_i\frac{\partial }{\partial y_i}{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}=-\sum _{i=\mathrm{1}}^r\left(n_i-m_i+\mathrm{1}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\end{array}$$ (32)

for n _i ≥ m _i − 1, i = 1,2,…, r; n _j ≥ 0, j ≠ i.

Similarly, as a consequence of Theorem 7, we can give the next result at once.

Corollary 10 —

Other recurrence relations for the family of multivariable polynomials Θ_n1,…,nr ^(α)(x, y, m) are

$$\begin{array}{c}-m_i\frac{\partial }{\partial y_i}{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \hspace{1em}\hspace{1em}=\frac{\partial }{\partial x_i}{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i,n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\end{array}$$ (33)

for n _i ≥ m _i, i = 1,2,…, r; n _j ≥ 0, j ≠ i and

$$\begin{array}{c}\frac{\partial }{\partial y_i}{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)=\mathrm{0}\end{array}$$ (34)

for n _i = 1,2,…, m _i − 1, i = 1,2,…, r; n _j ≥ 0, j ≠ i.

Theorem 11 —

The generating relation (2) yields the following addition formula for the family of multivariable polynomials given by (3)

$$\begin{array}{c}{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha +\beta \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)=\sum _{k_{\mathrm{1}}=\mathrm{0}}^{n_{\mathrm{1}}}\cdots \sum _{k_r=\mathrm{0}}^{n_r}{\mathrm{\Theta }}_{n_{\mathrm{1}}-k_{\mathrm{1}},\dots ,n_r-k_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \times {\mathrm{\Theta }}_{k_{\mathrm{1}},\dots ,k_r}^{\left(\beta \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right).\end{array}$$ (35)

Proof —

From (2), we have

$$\begin{array}{c}\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{(\alpha +\beta )}(\mathbf{x},\mathbf{y},\mathbf{m})t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}\\ \hspace{1em}\hspace{1em}={\left(\mathrm{1}-\sum _{i=\mathrm{1}}^r\left(m_i{x}_i{t}_i-y_i{t}_i^{m_i}\right)\right)}^{-\left(\alpha +\beta \right)}\\ \hspace{1em}\hspace{1em}={\left(\mathrm{1}-\sum _{i=\mathrm{1}}^r\left(m_i{x}_i{t}_i-y_i{t}_i^{m_i}\right)\right)}^{-\alpha }\\ \hspace{1em}\hspace{1em}\hspace{1em}\times {\left(\mathrm{1}-\sum _{i=\mathrm{1}}^r\left(m_i{x}_i{t}_i-y_i{t}_i^{m_i}\right)\right)}^{-\beta }\\ \hspace{1em}\hspace{1em}=\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}(\mathbf{x},\mathbf{y},\mathbf{m})t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}\\ \hspace{1em}\hspace{1em}\hspace{1em}\times \sum _{k_{\mathrm{1}},\dots ,k_r=\mathrm{0}}^{\infty }{\mathrm{\Theta }}_{k_{\mathrm{1}},\dots ,k_r}^{\left(\beta \right)}(\mathbf{x},\mathbf{y},\mathbf{m})t_{\mathrm{1}}^{k_{\mathrm{1}}}\cdots t_r^{k_r}.\end{array}$$ (36)

Replacing n _i by n _i − k _i, i = 1,2,…, r,   the right-hand side of the last equality is

$$\begin{array}{c}\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }\hspace{0.17em}‍\sum _{k_{\mathrm{1}}=\mathrm{0}}^{n_{\mathrm{1}}}\cdots \sum _{k_r=\mathrm{0}}^{n_r}{\mathrm{\Theta }}_{n_{\mathrm{1}}-k_{\mathrm{1}},\dots ,n_r-k_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \times {\mathrm{\Theta }}_{k_{\mathrm{1}},\dots ,k_r}^{\left(\beta \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}.\end{array}$$ (37)

Comparing the coefficients of t ₁ ^n₁ ⋯ t _r ^n_r completes the proof.

We now give expansions of the family of multivariable polynomials Θ_n1,…,nr ^(α)(x, y, m) given explicitly by (3) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials.

Theorem 12 —

Expansions of Θ_n1,…,nr ^(α)(x, y, m) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are as follows

$$\begin{array}{c}{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \hspace{1em}=\prod _{i=\mathrm{1}}^r\{\sum _{k_i=\mathrm{0}}^{\left[n_i/m_i\right]}\hspace{0.17em}\sum _{s_i=\mathrm{0}}^{\left[\left(n_i-m_i{k}_i\right)/\mathrm{2}\right]}\frac{\left(\mathrm{2}{n}_i-\mathrm{2}{m}_i{k}_i-\mathrm{4}{s}_i+\mathrm{1}\right)}{k_i!s_i!{\left(\mathrm{3}/\mathrm{2}\right)}_{n_i-m_i{k}_i-s_i}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left(-\mathrm{1}\right)}^{k_i}{y}_i^{k_i}{P}_{n_i-m_i{k}_i-\mathrm{2}{s}_i}\left(\frac{m_i{x}_i}{\mathrm{2}}\right)\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r-\left(m_{\mathrm{1}}-\mathrm{1}\right)k_{\mathrm{1}}-\cdots -\left(m_r-\mathrm{1}\right)k_r},\\ {\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \hspace{1em}=\prod _{i=\mathrm{1}}^r\{\sum _{k_i=\mathrm{0}}^{\left[n_i/m_i\right]}\hspace{0.17em}\sum _{s_i=\mathrm{0}}^{\left[\left(n_i-m_i{k}_i\right)/\mathrm{2}\right]}\frac{\left({\nu }_i+n_i-m_i{k}_i-\mathrm{2}{s}_i\right)}{k_i!s_i!{\left({\nu }_i\right)}_{n_i-m_i{k}_i-s_i+\mathrm{1}}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left(-y_i\right)}^{k_i}{C}_{n_i-m_i{k}_i-\mathrm{2}{s}_i}^{{\nu }_i}\left(\frac{m_i{x}_i}{\mathrm{2}}\right)\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\times {\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r-\left(m_{\mathrm{1}}-\mathrm{1}\right)k_{\mathrm{1}}-\cdots -\left(m_r-\mathrm{1}\right)k_r},\\ {\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \hspace{1em}=\prod _{i=\mathrm{1}}^r\left\{\sum _{k_i=\mathrm{0}}^{\left[n_i/m_i\right]}\hspace{0.17em}\sum _{s_i=\mathrm{0}}^{\left[\left(n_i-m_i{k}_i\right)/\mathrm{2}\right]}\frac{{\left(-y_i\right)}^{k_i}{H}_{n_i-m_i{k}_i-\mathrm{2}{s}_i}\left(m_i{x}_i/\mathrm{2}\right)}{k_i!s_i!\left(n_i-m_i{k}_i-\mathrm{2}{s}_i\right)!}\right\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\times {\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r-\left(m_{\mathrm{1}}-\mathrm{1}\right)k_{\mathrm{1}}-\cdots -\left(m_r-\mathrm{1}\right)k_r},\\ {\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)\\ \hspace{1em}=\prod _{i=\mathrm{1}}^r\{\sum _{k_i=\mathrm{0}}^{\left[n_i/m_i\right]}\hspace{0.17em}‍\sum _{s_i=\mathrm{0}}^{n_i-m_i{k}_i}\frac{{\left({\beta }_i+\mathrm{1}\right)}_{n_i-m_i{k}_i}{\mathrm{2}}^{n_i-m_i{k}_i}{\left(-\mathrm{1}\right)}^{s_i}}{k_i!\left(n_i-m_i{k}_i-s_i\right)!{\left({\beta }_i+\mathrm{1}\right)}_{s_i}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left(-y_i\right)}^{k_i}{L}_{s_i}^{\left({\beta }_i\right)}\left(\frac{m_i{x}_i}{\mathrm{2}}\right)\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r-\left(m_{\mathrm{1}}-\mathrm{1}\right)k_{\mathrm{1}}-\cdots -\left(m_r-\mathrm{1}\right)k_r}.\end{array}$$ (38)

Proof —

By (2), we get

$$\begin{array}{c}\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}\\ \hspace{1em}={\left(\mathrm{1}-\sum _{i=\mathrm{1}}^r\left(m_i{x}_i{t}_i-y_i{t}_i^{m_i}\right)\right)}^{-\alpha }\\ \hspace{1em}=\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }\hspace{0.17em}‍\sum _{k_{\mathrm{1}},\dots ,k_r=\mathrm{0}}^{\infty }\frac{{\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r+k_{\mathrm{1}}+\cdots +k_r}}{k_{\mathrm{1}}!\cdots k_r!n_{\mathrm{1}}!\cdots n_r!}{\left(-\mathrm{1}\right)}^{k_{\mathrm{1}}+\cdots +k_r}\\ \times {\left(m_{\mathrm{1}}{x}_{\mathrm{1}}\right)}^{n_{\mathrm{1}}}\cdots {\left(m_r{x}_r\right)}^{n_r}{y}_{\mathrm{1}}^{k_{\mathrm{1}}}\cdots y_r^{k_r}\\ \times t_{\mathrm{1}}^{n_{\mathrm{1}}+m_{\mathrm{1}}{k}_{\mathrm{1}}}\cdots t_r^{n_r+m_r{k}_r}.\end{array}$$ (39)

If we use the result in [11, page 181]

$$\begin{array}{c}\frac{{\left(mx\right)}^n}{n!}=\sum _{s=\mathrm{0}}^{\left[n/\mathrm{2}\right]}\frac{\left(\mathrm{2}n-\mathrm{4}s+\mathrm{1}\right)P_{n-\mathrm{2}s}\left(mx/\mathrm{2}\right)}{s!{\left(\mathrm{3}/\mathrm{2}\right)}_{n-s}},\end{array}$$ (40)

we can write that

$$\begin{array}{c}\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}\\ \hspace{1em}=\prod _{i=\mathrm{1}}^r\{\sum _{n_i=\mathrm{0}}^{\infty }\hspace{0.17em}‍\sum _{k_i=\mathrm{0}}^{\infty }\hspace{0.17em}‍\sum _{s_i=\mathrm{0}}^{\left[n_i/\mathrm{2}\right]}\frac{\left(\mathrm{2}{n}_i-\mathrm{4}{s}_i+\mathrm{1}\right)}{s_i!k_i!{\left(\mathrm{3}/\mathrm{2}\right)}_{n_i-s_i}}{P}_{n_i-\mathrm{2}{s}_i}\left(\frac{m_i{x}_i}{\mathrm{2}}\right)\\ \times {\left(-y_i\right)}^{k_i}{t}_i^{n_i+m_i{k}_i}\}{\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r+k_{\mathrm{1}}+\cdots +k_r}.\end{array}$$ (41)

Getting n _i − m _i k _i instead of n _i in the last equality, we have

$$\begin{array}{c}\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{\mathrm{\Theta }}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathbf{x},\mathbf{y},\mathbf{m}\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}\\ \hspace{1em}=\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }\hspace{0.17em}‍\prod _{i=\mathrm{1}}^r\{\sum _{k_i=\mathrm{0}}^{\left[n_i/m_i\right]}\hspace{0.17em}\sum _{s_i=\mathrm{0}}^{\left[\left(n_i-m_i{k}_i\right)/\mathrm{2}\right]}\frac{\left(\mathrm{2}{n}_i-\mathrm{2}{m}_i{k}_i-\mathrm{4}{s}_i+\mathrm{1}\right)}{s_i!k_i!{\left(\mathrm{3}/\mathrm{2}\right)}_{n_i-m_i{k}_i-s_i}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times P_{n_i-m_i{k}_i-\mathrm{2}{s}_i}\left(\frac{m_i{x}_i}{\mathrm{2}}\right){\left(-y_i\right)}^{k_i}{t}_i^{n_i}\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r-\left(m_{\mathrm{1}}-\mathrm{1}\right)k_{\mathrm{1}}-\cdots -\left(m_r-\mathrm{1}\right)k_r}\\ \hspace{1em}=\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }\{\prod _{i=\mathrm{1}}^r\sum _{k_i=\mathrm{0}}^{\left[n_i/m_i\right]}\hspace{0.17em}\sum _{s_i=\mathrm{0}}^{\left[\left(n_i-m_i{k}_i\right)/\mathrm{2}\right]}\frac{\left(\mathrm{2}{n}_i-\mathrm{2}{m}_i{k}_i-\mathrm{4}{s}_i+\mathrm{1}\right)}{s_i!k_i!{\left(\mathrm{3}/\mathrm{2}\right)}_{n_i-m_i{k}_i-s_i}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times P_{n_i-m_i{k}_i-\mathrm{2}{s}_i}\left(\frac{m_i{x}_i}{\mathrm{2}}\right){\left(-y_i\right)}^{k_i}\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r-\left(m_{\mathrm{1}}-\mathrm{1}\right)k_{\mathrm{1}}-\cdots -\left(m_r-\mathrm{1}\right)k_r}{t}_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}.\end{array}$$ (42)

Comparing the coefficients of t ₁ ^n₁ ⋯ t _r ^n_r gives the desired relation.

In a similar manner, in (39), using the following results, respectively, [11, page 283 (36), page 194 (4), page 207 (2)]

$$\begin{array}{c}\frac{{\left(mx\right)}^n}{n!}=\sum _{k=\mathrm{0}}^{\left[n/\mathrm{2}\right]}\frac{\left(\nu +n-\mathrm{2}k\right)C_{n-\mathrm{2}k}^{\nu }\left(mx/\mathrm{2}\right)}{k!{\left(\nu \right)}_{n+\mathrm{1}-k}},\\ \frac{{\left(mx\right)}^n}{n!}=\sum _{k=\mathrm{0}}^{\left[n/\mathrm{2}\right]}\frac{H_{n-\mathrm{2}k}\left(mx/\mathrm{2}\right)}{k!\left(n-\mathrm{2}k\right)!},\\ \frac{{\left(mx\right)}^n}{n!}={\mathrm{2}}^n\sum _{k=\mathrm{0}}^n\frac{{\left(-\mathrm{1}\right)}^k{\left(\alpha +\mathrm{1}\right)}_n{L}_k^{\left(\alpha \right)}\left(mx/\mathrm{2}\right)}{\left(n-k\right)!{\left(\alpha +\mathrm{1}\right)}_k},\end{array}$$ (43)

one can easily obtain the other expansions of Θ_n1,…,nr ^(α)(x, y, m) in series of Gegenbauer, Hermite, and Laguerre polynomials.

5. The Special Cases of Θ_n ^(α) (x, y, m) and Some Properties

In this section, we discuss some special cases of the family of multivariable polynomials Θ_n ^(α)(x, y, m) and give their several properties.

5.1. The Case of m _i = 2, y _i = 1, i = 1,2,…, r in (2)

This case gives a multivariable analogue of Gegenbauer polynomials

$$\begin{array}{c}{\left(\mathrm{1}-\mathrm{2}{x}_{\mathrm{1}}{t}_{\mathrm{1}}+t_{\mathrm{1}}^{\mathrm{2}}-\cdots -\mathrm{2}{x}_r{t}_r+t_r^{\mathrm{2}}\right)}^{-\alpha }\\ \hspace{1em}\hspace{1em}=\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r},\end{array}$$ (44)

which reduces to two variables Gegenbauer polynomials given by [12] for r = 2. Equation (44) yields the following formula:

$$\begin{array}{c}{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}=\sum _{k_{\mathrm{1}}=\mathrm{0}}^{\left[n_{\mathrm{1}}/\mathrm{2}\right]}\cdots \sum _{k_r=\mathrm{0}}^{\left[n_r/\mathrm{2}\right]}(\left({\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r-k_{\mathrm{1}}-\cdots -k_r}\right)\\ \times (k_{\mathrm{1}}!\cdots k_r!\left(n_{\mathrm{1}}-\mathrm{2}{k}_{\mathrm{1}}\right)!\\ {\cdots \left(n_r-\mathrm{2}{k}_r\right)!)}^{-1})\\ \times {\left(-\mathrm{1}\right)}^{k_{\mathrm{1}}+\cdots +k_r}{\left(\mathrm{2}{x}_{\mathrm{1}}\right)}^{n_{\mathrm{1}}-\mathrm{2}{k}_{\mathrm{1}}}\cdots {\left(\mathrm{2}{x}_r\right)}^{n_r-\mathrm{2}{k}_r}.\end{array}$$ (45)

It follows that C _n1,…,nr ^(α)(x ₁,…, x _r) is a polynomial of degree n _i with respect to the fixed variable x _i  (i = 1,2,…, r).   Thus, C _n1,…,nr ^(α)(x ₁,…, x _r) is a polynomial of total degree (n ₁ + ⋯+n _r) with respect to the variables x ₁,…, x _r. Equation (45) also yields

$$\begin{array}{c}{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}=\frac{{\mathrm{2}}^{n_{\mathrm{1}}+\cdots +n_r}{\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r}}{n_{\mathrm{1}}!\cdots n_r!}{x}_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots x_r^{n_r}+\Pi \left(x_{\mathrm{1}},\dots ,x_r\right),\end{array}$$ (46)

where Π(x ₁,…, x _r) is a polynomial of degree (n ₁ + ⋯+n _r − 2) with respect to the variables x ₁,…, x _r. In (44), by getting x ₁ → −x ₁ and t ₁ → −t ₁, we have

$$\begin{array}{c}{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(-x_{\mathrm{1}},x_{\mathrm{2}},\dots ,x_r\right)={\left(-\mathrm{1}\right)}^{n_{\mathrm{1}}}{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right).\end{array}$$ (47)

Similarly, for i = 1,2,…, r,   we get

$$\begin{array}{c}{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_{i-\mathrm{1}},-x_i,x_{i+\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}={\left(-\mathrm{1}\right)}^{n_i}{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right).\end{array}$$ (48)

Taking x _i → −x _i and  t _i → −t _i, i = 1,2,…, r, in (44), we obtain

$$\begin{array}{c}{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(-x_{\mathrm{1}},\dots ,-x_r\right)={\left(-\mathrm{1}\right)}^{n_{\mathrm{1}}+\cdots +n_r}{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right).\end{array}$$ (49)

Theorem 13 —

For the polynomials C _n1,…,nr ^(α)(x ₁,…, x _r), one has

$$\begin{array}{c}{C}_{\mathrm{2}{n}_{\mathrm{1}},\dots ,\mathrm{2}{n}_r}^{\left(\alpha \right)}\left(\mathrm{0,0},\dots ,\mathrm{0}\right)=\frac{{\left(-\mathrm{1}\right)}^{n_{\mathrm{1}}+\cdots +n_r}{\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r}}{n_{\mathrm{1}}!\cdots n_r!}\end{array}$$ (50)

and if at least one of n _i, i = 1,2,…, r, is odd; then

$$\begin{array}{c}{C}_{n_{\mathrm{1}},\dots .,n_r}^{\left(\alpha \right)}\left(\mathrm{0,0},\dots ,\mathrm{0}\right)=\mathrm{0}.\end{array}$$ (51)

Proof —

If we set all x _i = 0, i = 1,2,…, r in (44), we have

$$\begin{array}{c}{\left(\mathrm{1}+t_{\mathrm{1}}^{\mathrm{2}}+\cdots +t_r^{\mathrm{2}}\right)}^{-\alpha }=\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(\mathrm{0,0},\dots ,\mathrm{0}\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}.\end{array}$$ (52)

On the other hand, we get

$$\begin{array}{c}{\left(\mathrm{1}+t_{\mathrm{1}}^{\mathrm{2}}+\cdots +t_r^{\mathrm{2}}\right)}^{-\alpha }\\ \hspace{1em}=\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }\frac{{\left(-\mathrm{1}\right)}^{n_{\mathrm{1}}+\cdots +n_r}}{n_{\mathrm{1}}!\cdots n_r!}{\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r}{t}_{\mathrm{1}}^{\mathrm{2}{n}_{\mathrm{1}}}\cdots t_r^{\mathrm{2}{n}_r}.\end{array}$$ (53)

By comparing the coefficients of t ₁ ^n₁ ⋯ t _r ^n_r, we obtain the desired.

From the theorems and corollaries given in Section 4, we can give some other properties of C _n1,…,nr ^(α)(x ₁,…, x _r).

Remark 14 —

By Corollaries 8 and 9, for the family of multivariable polynomials generated by (44), the following relations:

$$\begin{array}{c}{n}_i{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}(x_{\mathrm{1}},\dots ,x_r)\\ \hspace{1em}\hspace{1em}=x_i\frac{\partial }{\partial x_i}{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}(x_{\mathrm{1}},\dots ,x_r)\\ \hspace{1em}\hspace{1em}\hspace{1em}-\frac{\partial }{\partial x_i}{C}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right),\\ \sum _{i=\mathrm{1}}^r{n}_i{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}(x_{\mathrm{1}},\dots ,x_r)\\ \hspace{1em}\hspace{1em}=\sum _{i=\mathrm{1}}^r{x}_i\frac{\partial }{\partial x_i}{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}(x_{\mathrm{1}},\dots ,x_r)\\ \hspace{1em}\hspace{1em}\hspace{1em}-\sum _{i=\mathrm{1}}^r\frac{\partial }{\partial x_i}{C}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}(x_{\mathrm{1}},\dots ,x_r)\end{array}$$ (54)

hold for n _i ≥ 1,  i = 1,2,…, r.

Remark 15 —

From Theorem 11, the multivariable polynomials C _n1,…,nr ^(α)(x ₁,…, x _r) satisfy the following addition formula:

$$\begin{array}{c}{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha +\beta \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)=\sum _{k_{\mathrm{1}}=\mathrm{0}}^{n_{\mathrm{1}}}\cdots \sum _{k_r=\mathrm{0}}^{n_r}{C}_{n_{\mathrm{1}}-k_{\mathrm{1}},\dots ,n_r-k_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times C_{k_{\mathrm{1}},\dots ,k_r}^{\left(\beta \right)}\left(x_{\mathrm{1}},\dots ,x_r\right).\end{array}$$ (55)

Remark 16 —

As a result of Theorem 12, expansions of C _n1,…,nr ^(α)(x ₁,…, x _r) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are as follows:

$$\begin{array}{c}{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}=\prod _{i=\mathrm{1}}^r\{\sum _{k_i=\mathrm{0}}^{\left[n_i/\mathrm{2}\right]}\sum _{s_i=\mathrm{0}}^{\left[\left(n_i-\mathrm{2}{k}_i\right)/\mathrm{2}\right]}\frac{\left(\mathrm{2}{n}_i-\mathrm{4}{k}_i-\mathrm{4}{s}_i+\mathrm{1}\right)}{k_i!s_i!{\left(\mathrm{3}/\mathrm{2}\right)}_{n_i-\mathrm{2}{k}_i-s_i}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\times {\left(-\mathrm{1}\right)}^{k_i}{P}_{n_i-\mathrm{2}{k}_i-\mathrm{2}{s}_i}\left(x_i\right)\}\\ {\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r-k_{\mathrm{1}}-\cdots -k_r},\\ C_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}=\prod _{i=\mathrm{1}}^r\{\sum _{k_i=\mathrm{0}}^{\left[n_i/\mathrm{2}\right]}\sum _{s_i=\mathrm{0}}^{\left[\left(n_i-\mathrm{2}{k}_i\right)/\mathrm{2}\right]}\frac{\left({\nu }_i+n_i-\mathrm{2}{k}_i-\mathrm{2}{s}_i\right)}{k_i!s_i!{\left({\nu }_i\right)}_{n_i-\mathrm{2}{k}_i-s_i+\mathrm{1}}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\times {\left(-\mathrm{1}\right)}^{k_i}{C}_{n_i-\mathrm{2}{k}_i-\mathrm{2}{s}_i}^{{\nu }_i}\left(x_i\right)\}\\ {\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r-k_{\mathrm{1}}-\cdots -k_r},\\ C_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}=\prod _{i=\mathrm{1}}^r\left\{\sum _{k_i=\mathrm{0}}^{\left[n_i/\mathrm{2}\right]}\sum _{s_i=\mathrm{0}}^{\left[\left(n_i-\mathrm{2}{k}_i\right)/\mathrm{2}\right]}\frac{{\left(-\mathrm{1}\right)}^{k_i}{H}_{n_i-\mathrm{2}{k}_i-\mathrm{2}{s}_i}\left(x_i\right)}{k_i!s_i!\left(n_i-\mathrm{2}{k}_i-\mathrm{2}{s}_i\right)!}\right\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r-k_{\mathrm{1}}-\cdots -k_r,}\\ C_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}=\prod _{i=\mathrm{1}}^r\{\sum _{k_i=\mathrm{0}}^{\left[n_i/\mathrm{2}\right]}\hspace{0.17em}\sum _{s_i=\mathrm{0}}^{n_i-\mathrm{2}{k}_i}\frac{{\left({\beta }_i+\mathrm{1}\right)}_{n_i-\mathrm{2}{k}_i}{\mathrm{2}}^{n_i-\mathrm{2}{k}_i}{\left(-\mathrm{1}\right)}^{s_i}}{k_i!\left(n_i-\mathrm{2}{k}_i-s_i\right)!{\left({\beta }_i+\mathrm{1}\right)}_{s_i}}\\ \times {\left(-\mathrm{1}\right)}^{k_i}{L}_{s_i}^{\left({\beta }_i\right)}\left(x_i\right)\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r-k_{\mathrm{1}}-\cdots -k_r}.\end{array}$$ (56)

We now give a hypergeometric representation for the multivariable polynomials C _n1,…,nr ^(α)(x ₁,…, x _r) given by (44).

Theorem 17 —

The multivariable polynomials C _n1,…,nr ^(α)(x ₁,…, x _r) have the following hypergeometric representation:

$$\begin{array}{c}{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}=\frac{{\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r}}{n_{\mathrm{1}}!\cdots n_r!}{\left(\mathrm{2}{x}_{\mathrm{1}}\right)}^{n_{\mathrm{1}}}\cdots {\left(\mathrm{2}{x}_r\right)}^{n_r}\\ \hspace{1em}\hspace{1em}\hspace{1em}\times F_B^{\left(r\right)}[-\frac{n_{\mathrm{1}}}{\mathrm{2}},\dots ,-\frac{n_r}{\mathrm{2}},\frac{-n_{\mathrm{1}}+\mathrm{1}}{\mathrm{2}},\dots ,\frac{-n_r+\mathrm{1}}{\mathrm{2}};\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{1}-\alpha -n_{\mathrm{1}}-\cdots -n_r;\frac{\mathrm{1}}{x_{\mathrm{1}}^{\mathrm{2}}},\dots ,\frac{\mathrm{1}}{x_r^{\mathrm{2}}}],\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\left(\max \left\{\frac{\mathrm{1}}{x_{\mathrm{1}}^{\mathrm{2}}},\dots ,\frac{\mathrm{1}}{x_r^{\mathrm{2}}}\right\}<\mathrm{1}\right),\end{array}$$ (57)

where F _B ^(r) is a Lauricella function of r-variable defined by [13] (see also [2])

$$\begin{array}{c}{F}_B^{\left(r\right)}\left[a_{\mathrm{1}},\dots ,a_r,b_{\mathrm{1}},\dots ,b_r;c;x_{\mathrm{1}},\dots ,x_r\right]\\ \hspace{1em}\hspace{1em}=\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }\frac{{\left(a_{\mathrm{1}}\right)}_{n_{\mathrm{1}}}\cdots {\left(a_r\right)}_{n_r}{\left(b_{\mathrm{1}}\right)}_{n_{\mathrm{1}}}\cdots {\left(b_r\right)}_{n_r}}{{\left(c\right)}_{n_{\mathrm{1}}+\cdots +n_r}}\frac{x_{\mathrm{1}}^{n_{\mathrm{1}}}}{n_{\mathrm{1}}!}\cdots \frac{x_r^{n_r}}{n_r!},\\ \left(\max \left\{\left|x_{\mathrm{1}}\right|,\dots ,\left|x_r\right|\right\}<\mathrm{1}\right).\end{array}$$ (58)

Proof —

We have the following results from [11]:

$$\begin{array}{c}{\left(\alpha \right)}_{n-k}=\frac{{\left(-\mathrm{1}\right)}^k{\left(\alpha \right)}_n}{{\left(\mathrm{1}-\alpha -n\right)}_k},\hspace{1em}\hspace{1em}{\left(-n\right)}_{\mathrm{2}k}={\mathrm{2}}^{\mathrm{2}k}{\left(-\frac{n}{\mathrm{2}}\right)}_k{\left(\frac{-n+\mathrm{1}}{\mathrm{2}}\right)}_k,\\ \left(n-\mathrm{2}k\right)!=\frac{n!}{{\left(-n\right)}_{\mathrm{2}k}}.\end{array}$$ (59)

In view of these relations, the equality (45) can be written as follows:

$$\begin{array}{c}{C}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}=\sum _{k_{\mathrm{1}}=\mathrm{0}}^{\left[n_{\mathrm{1}}/\mathrm{2}\right]}\cdots \sum _{k_r=\mathrm{0}}^{\left[n_r/\mathrm{2}\right]}(({\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r}{\mathrm{2}}^{\mathrm{2}\left(k_{\mathrm{1}}+\cdots +k_r\right)}{\left(-\frac{n_{\mathrm{1}}}{\mathrm{2}}\right)}_{k_{\mathrm{1}}}‍\\ \times {\left(\frac{-n_{\mathrm{1}}+\mathrm{1}}{\mathrm{2}}\right)}_{k_{\mathrm{1}}}\cdots {\left(-\frac{n_r}{\mathrm{2}}\right)}_{k_r}\\ {\times \left(\frac{-n_r+\mathrm{1}}{\mathrm{2}}\right)}_{k_r})\\ \times (k_{\mathrm{1}}!\cdots k_r!n_{\mathrm{1}}!\cdots n_r!\\ {{\times \left(\mathrm{1}-\alpha -n_{\mathrm{1}}-\cdots -n_r\right)}_{k_{\mathrm{1}}+\cdots +k_r})}^{-1})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left(\mathrm{2}{x}_{\mathrm{1}}\right)}^{n_{\mathrm{1}}-\mathrm{2}{k}_{\mathrm{1}}}\cdots {\left(\mathrm{2}{x}_r\right)}^{n_r-\mathrm{2}{k}_r}\\ \hspace{1em}=\frac{{\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r}}{n_{\mathrm{1}}!\cdots n_r!}{\left(\mathrm{2}{x}_{\mathrm{1}}\right)}^{n_{\mathrm{1}}}\cdots {\left(\mathrm{2}{x}_r\right)}^{n_r}\\ \hspace{1em}\hspace{1em}\times \sum _{k_{\mathrm{1}}=\mathrm{0}}^{\left[n_{\mathrm{1}}/\mathrm{2}\right]}\cdots \sum _{k_r=\mathrm{0}}^{\left[n_r/\mathrm{2}\right]}(({\left(-\frac{n_{\mathrm{1}}}{\mathrm{2}}\right)}_{k_{\mathrm{1}}}{\left(\frac{-n_{\mathrm{1}}+\mathrm{1}}{\mathrm{2}}\right)}_{k_{\mathrm{1}}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\cdots {\left(-\frac{n_r}{\mathrm{2}}\right)}_{k_r}{\left(\frac{-n_r+\mathrm{1}}{\mathrm{2}}\right)}_{k_r})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times ({\left(\mathrm{1}-\alpha -n_{\mathrm{1}}-\cdots -n_r\right)}_{k_{\mathrm{1}}+\cdots +k_r}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {k_{\mathrm{1}}!\cdots k_r!)}^{-1})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\hspace{0.5em}\times {\left(\frac{\mathrm{1}}{x_{\mathrm{1}}^{\mathrm{2}}}\right)}^{k_{\mathrm{1}}}\cdots {\left(\frac{\mathrm{1}}{x_r^{\mathrm{2}}}\right)}^{k_r}\\ \hspace{1em}=F_B^{\left(r\right)}[-\frac{n_{\mathrm{1}}}{\mathrm{2}},\dots ,-\frac{n_r}{\mathrm{2}},\frac{-n_{\mathrm{1}}+\mathrm{1}}{\mathrm{2}},\dots ,\frac{-n_r+\mathrm{1}}{\mathrm{2}};\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{1}-\alpha -n_{\mathrm{1}}-\cdots -n_r;\frac{\mathrm{1}}{x_{\mathrm{1}}^{\mathrm{2}}},\dots ,\frac{\mathrm{1}}{x_r^{\mathrm{2}}}]\\ \hspace{1em}\hspace{1em}\times \frac{{\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r}}{n_{\mathrm{1}}!\cdots n_r!}{\left(\mathrm{2}{x}_{\mathrm{1}}\right)}^{n_{\mathrm{1}}}\cdots {\left(\mathrm{2}{x}_r\right)}^{n_r},\end{array}$$ (60)

where max⁡{1/x ₁ ²,…, 1/x _r ²} < 1. The proof is completed.

Remark 18 —

For r = 2, Theorem 17 reduces to the known result for two variable Gegenbauer polynomials given by [12].

5.2. The Case of x _i = 0, y _i = −x _i, i = 1,2,…, r in (2)

This case yields

$$\begin{array}{c}{\left(\mathrm{1}-x_{\mathrm{1}}{t}_{\mathrm{1}}^{m_{\mathrm{1}}}-x_{\mathrm{2}}{t}_{\mathrm{2}}^{m_{\mathrm{2}}}-\cdots -x_r{t}_r^{m_r}\right)}^{-\alpha }\\ \hspace{1em}\hspace{1em}=\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{U}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r},\end{array}$$ (61)

which is a different unification from that in [8].

Remark 19 —

From Corollaries 8 and 9, the multivariable polynomials U _n1,…,nr ^(α)(x ₁,…, x _r) satisfy

$$\begin{array}{c}{m}_i{x}_i\frac{\partial }{\partial x_i}{U}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}=\left(n_i-m_i+\mathrm{1}\right)U_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right),\\ \sum _{i=\mathrm{1}}^r{m}_i{x}_i\frac{\partial }{\partial x_i}\\ \hspace{1em}\hspace{1em}\times U_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}=\sum _{i=\mathrm{1}}^r\left(n_i-m_i+\mathrm{1}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times U_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-m_i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right).\end{array}$$ (62)

for n _i ≥ m _i − 1, i = 1,2,…, r; n _j ≥ 0, j ≠ i.

Remark 20 —

As result of Theorem 11, the multivariable polynomials U _n1,…,nr ^(α)(x ₁,…, x _r) have the following addition formula:

$$\begin{array}{c}{U}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha +\beta \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)=\sum _{k_{\mathrm{1}}=\mathrm{0}}^{n_{\mathrm{1}}}\cdots \sum _{k_r=\mathrm{0}}^{n_r}{U}_{n_{\mathrm{1}}-k_{\mathrm{1}},\dots ,n_r-k_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times U_{k_{\mathrm{1}},\dots ,k_r}^{\left(\beta \right)}\left(x_{\mathrm{1}},\dots ,x_r\right).\end{array}$$ (63)

5.2.1. The Case of m _i = 1, i = 1,2,…, r in Section 5.2

In this case, we get a r-variable analogue of Lagrange polynomials (see [14]) which is different from that in [10]

$$\begin{array}{c}{\left(\mathrm{1}-x_{\mathrm{1}}{t}_{\mathrm{1}}-\cdots -x_r{t}_r\right)}^{-\alpha }\\ \hspace{1em}\hspace{1em}=\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{G}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r}.\end{array}$$ (64)

They are given explicitly by

$$\begin{array}{c}{G}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)=\frac{{\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r}}{n_{\mathrm{1}}!\cdots n_r!}{x}_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots x_r^{n_r}.\end{array}$$ (65)

Remark 21 —

From Corollaries 8 and 9, the multivariable polynomials G _n1,…,nr ^(α)(x ₁,…, x _r) satisfy

$$\begin{array}{c}{n}_i{G}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)=x_i\frac{\partial }{\partial x_i}{G}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\end{array}$$ (66)

for i = 1,2,…, r and

$$\begin{array}{c}\sum _{i=\mathrm{1}}^r{n}_i{G}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}=\sum _{i=\mathrm{1}}^r{x}_i\frac{\partial }{\partial x_i}{G}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right).\end{array}$$ (67)

Remark 22 —

By Theorem 11, the multivariable polynomials G _n1,…,nr ^(α)(x ₁,…, x _r) have the following addition formula:

$$\begin{array}{c}{G}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha +\beta \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)=\sum _{k_{\mathrm{1}}=\mathrm{0}}^{n_{\mathrm{1}}}\cdots \sum _{k_r=\mathrm{0}}^{n_r}{G}_{n_{\mathrm{1}}-k_{\mathrm{1}},\dots ,n_r-k_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \times G_{k_{\mathrm{1}},\dots ,k_r}^{\left(\beta \right)}\left(x_{\mathrm{1}},\dots ,x_r\right).\end{array}$$ (68)

Remark 23 —

From Theorem 12, expansions of G _n1,…,nr ^(α)(x ₁,…, x _r) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are given by

$$\begin{array}{c}{G}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}=\prod _{i=\mathrm{1}}^r\left\{\sum _{s_i=\mathrm{0}}^{\left[n_i/\mathrm{2}\right]}\frac{\left(\mathrm{2}{n}_i-\mathrm{4}{s}_i+\mathrm{1}\right)}{s_i!{\left(\mathrm{3}/\mathrm{2}\right)}_{n_i-s_i}}{P}_{n_i-\mathrm{2}{s}_i}\left(\frac{x_i}{\mathrm{2}}\right)\right\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r},\\ G_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}=\prod _{i=\mathrm{1}}^r\left\{\sum _{s_i=\mathrm{0}}^{\left[n_i/\mathrm{2}\right]}\frac{\left({\nu }_i+n_i-\mathrm{2}{s}_i\right)}{s_i!{\left({\nu }_i\right)}_{n_i-s_i+\mathrm{1}}}{C}_{n_i-\mathrm{2}{s}_i}^{{\nu }_i}\left(\frac{x_i}{\mathrm{2}}\right)\right\}\\ {\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r},\\ G_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}=\prod _{i=\mathrm{1}}^r\left\{\sum _{s_i=\mathrm{0}}^{\left[n_i/\mathrm{2}\right]}\frac{H_{n_i-\mathrm{2}{s}_i}\left(x_i/\mathrm{2}\right)}{s_i!\left(n_i-\mathrm{2}{s}_i\right)!}\right\}{\left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r},\\ G_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}=\prod _{i=\mathrm{1}}^r\left\{\sum _{s_i=\mathrm{0}}^{n_i}\frac{{\left({\beta }_i+\mathrm{1}\right)}_{n_i}{\mathrm{2}}^{n_i}{\left(-\mathrm{1}\right)}^{s_i}}{\left(n_i-s_i\right)!{\left({\beta }_i+\mathrm{1}\right)}_{s_i}}{L}_{s_i}^{\left({\beta }_i\right)}\left(\frac{x_i}{\mathrm{2}}\right)\right\}\\ {\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \left(\alpha \right)}_{n_{\mathrm{1}}+\cdots +n_r}.\end{array}$$ (69)

We can discuss some generating functions for the multivariable polynomials G _n1,…,nr ^(α)(x ₁,…, x _r).

Theorem 24 —

For the polynomials G _n1,…,nr ^(α)(x ₁,…, x _r),   the following generating function holds true for k ∈ ℕ ₀:

$$\begin{array}{c}\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{G}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right){\left[k+n_i\right]}_m{z}_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots z_r^{n_r}\\ \hspace{1em}\hspace{1em}=m!{\left(\mathrm{1}-x_{\mathrm{1}}{z}_{\mathrm{1}}-\cdots -x_r{z}_r\right)}^{-\alpha }\hspace{0.17em}\\ \hspace{1em}\hspace{1em}\hspace{1em}\times {\text{g}\hspace{0.17em}}_m^{\left(\alpha ,-k\right)}\left(\frac{x_i{z}_i}{\mathrm{1}-x_{\mathrm{1}}{z}_{\mathrm{1}}-\cdots -x_r{z}_r},-\mathrm{1}\right),\end{array}$$ (70)

for each i = 1,2,…, r, where

$$\begin{array}{c}{\left[k\right]}_m:=k\left(k-\mathrm{1}\right)\cdots \left(k-m+\mathrm{1}\right),\hspace{1em}m\in \mathbb{N},{\left[k\right]}_{\mathrm{0}}=\mathrm{1}\end{array}$$ (71)

and   g  _n ^(α,β)(x, y) denotes the classical Lagrange polynomials defined by [14]

$$\begin{array}{c}{\left(\mathrm{1}-xt\right)}^{-\alpha }{\left(\mathrm{1}-yt\right)}^{-\beta }=\sum _{n=\mathrm{0}}^{\infty }{\hspace{0.17em}\text{g}\hspace{0.17em}}_n^{\left(\alpha ,\beta \right)}\left(x,y\right)t^n\\ \left(\alpha ,\beta \in ℂ;\left|t\right|<\min \left\{{\left|x\right|}^{-\mathrm{1}},{\left|y\right|}^{-\mathrm{1}}\right\}\right).\end{array}$$ (72)

Proof —

Fix i = 1,2,…, r. Let 𝒞 _ξiand 𝒞 _ξi* denote the circles of radius ɛ _i, centered ξ _i = z _i and ξ _i = 0, respectively, where

$$\begin{array}{c}\mathrm{0}<{\varepsilon }_i<{\left|x_i\right|}^{-\mathrm{1}}\left|\mathrm{1}-\sum _{k=\mathrm{1}}^r{x}_k{z}_k\right|.\end{array}$$ (73)

Here these circles are described in the positive direction. By Cauchy's Integral Formula, we have

$$\begin{array}{c}{D}_{z_i}^m\left\{{\left(\mathrm{1}-x_{\mathrm{1}}{z}_{\mathrm{1}}-\cdots -x_r{z}_r\right)}^{-\alpha }{z}_i^k\right\}\\ \hspace{1em}=\frac{m!}{\mathrm{2}\pi i}{\oint }_{{𝒞}_{{\xi }_i}}^{}\frac{{\xi }_i^k{\left(\mathrm{1}-x_{\mathrm{1}}{z}_{\mathrm{1}}-\cdots -x_i{\xi }_i-\cdots -x_r{z}_r\right)}^{-\alpha }}{{\left({\xi }_i-z_i\right)}^{m+\mathrm{1}}}d{\xi }_i\\ \hspace{1em}=\frac{m!}{\mathrm{2}\pi i}\\ \hspace{1em}\hspace{1em}\times {\oint }_{{𝒞}_{{\xi }_i}^{\ast }}^{}\frac{{\left({\xi }_i+z_i\right)}^k{\left(\mathrm{1}-x_{\mathrm{1}}{z}_{\mathrm{1}}-\cdots -x_i\left({\xi }_i+z_i\right)-\cdots -x_r{z}_r\right)}^{-\alpha }}{{\xi }_i^{m+\mathrm{1}}}d{\xi }_i\\ \hspace{1em}=\frac{m!}{\mathrm{2}\pi i}{z}_i^k{\left(\mathrm{1}-x_{\mathrm{1}}{z}_{\mathrm{1}}-\cdots -x_r{z}_r\right)}^{-\alpha }\\ \hspace{1em}\hspace{1em}\times {\oint }_{{𝒞}_{{\xi }_i}^{\ast }}^{}\frac{{\left(\mathrm{1}+{\xi }_i/z_i\right)}^k}{{\xi }_i^{m+\mathrm{1}}}{\left(\mathrm{1}-\frac{x_i{\xi }_i}{\mathrm{1}-x_{\mathrm{1}}{z}_{\mathrm{1}}-\cdots -x_r{z}_r}\right)}^{-\alpha }d{\xi }_i\\ \hspace{1em}=m!z_i^k{\left(\mathrm{1}-x_{\mathrm{1}}{z}_{\mathrm{1}}-\cdots -x_r{z}_r\right)}^{-\alpha }\\ \hspace{1em}\hspace{1em}\times {\text{g}}_m^{\left(\alpha ,-k\right)}\left(\frac{x_i}{\mathrm{1}-x_{\mathrm{1}}{z}_{\mathrm{1}}-\cdots -x_r{z}_r},-\frac{\mathrm{1}}{z_i}\right)\\ \hspace{1em}=m!z_i^{k-m}{\left(\mathrm{1}-x_{\mathrm{1}}{z}_{\mathrm{1}}-\cdots -x_r{z}_r\right)}^{-\alpha }\\ \hspace{1em}\hspace{1em}\times {\text{g}}_m^{\left(\alpha ,-k\right)}\left(\frac{x_i{z}_i}{\mathrm{1}-x_{\mathrm{1}}{z}_{\mathrm{1}}-\cdots -x_r{z}_r},-\mathrm{1}\right).\end{array}$$ (74)

On the other hand, we can write that

$$\begin{array}{c}{D}_{z_i}^m\left\{{\left(\mathrm{1}-x_{\mathrm{1}}{z}_{\mathrm{1}}-\cdots -x_r{z}_r\right)}^{-\alpha }{z}_i^k\right\}\\ \hspace{1em}=D_{z_i}^m\left\{z_i^k\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{G}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)z_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots z_r^{n_r}\right\}\\ \hspace{1em}=\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{G}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\frac{d^m}{d{z}_i^m}{z}_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots z_i^{n_i+k}\cdots z_r^{n_r}\\ \hspace{1em}=z_i^{k-m}\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{G}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right){\left[k+n_i\right]}_m{z}_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots z_r^{n_r}.\end{array}$$ (75)

From (74) and (75), we see that

$$\begin{array}{c}\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{G}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}(x_{\mathrm{1}},\dots ,x_r){\left[k+n_i\right]}_m{z}_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots z_r^{n_r}\\ \hspace{1em}=m!{\left(\mathrm{1}-x_{\mathrm{1}}{z}_{\mathrm{1}}-\cdots -x_r{z}_r\right)}^{-\alpha }\\ \hspace{1em}\hspace{1em}\times {\text{g}}_m^{\left(\alpha ,-k\right)}\left(\frac{x_i{z}_i}{\mathrm{1}-x_{\mathrm{1}}{z}_{\mathrm{1}}-\cdots -x_r{z}_r},-\mathrm{1}\right).\end{array}$$ (76)

Corollary 25 —

From (71), one observes that

$$\begin{array}{c}{\left[n_i+m\right]}_m=\left(n_i+m\right)\left(n_i+m-\mathrm{1}\right)\cdots \left(n_i+\mathrm{1}\right)\\ ={\left(n_i+\mathrm{1}\right)}_m\end{array}$$ (77)

for each i = 1,2,…, r. By setting k = m  in (70) and then using (77), one has

$$\begin{array}{c}\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{\left(n_i+\mathrm{1}\right)}_m{G}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)z_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots z_r^{n_r}\\ \hspace{1em}=m!{\left(\mathrm{1}-x_{\mathrm{1}}{z}_{\mathrm{1}}-\cdots -x_r{z}_r\right)}^{-\alpha }\\ \hspace{1em}\hspace{1em}\times {\text{g}\hspace{0.17em}}_m^{\left(\alpha ,-m\right)}\left(\frac{x_i{z}_i}{\mathrm{1}-x_{\mathrm{1}}{z}_{\mathrm{1}}-\cdots -x_r{z}_r},-\mathrm{1}\right).\end{array}$$ (78)

5.2.2. The Case of m _i  =  i,   i  =  1,2,…, r in Section 5.2

This case reduces to a multivariable analogue of Lagrange-Hermite polynomials

$$\begin{array}{c}{\left(\mathrm{1}-x_{\mathrm{1}}{t}_{\mathrm{1}}-x_{\mathrm{2}}{t}_{\mathrm{2}}^{\mathrm{2}}-\cdots -x_r{t}_r^r\right)}^{-\alpha }\\ \hspace{1em}\hspace{1em}=\sum _{n_{\mathrm{1}},\dots ,n_r=\mathrm{0}}^{\infty }{H}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)t_{\mathrm{1}}^{n_{\mathrm{1}}}\cdots t_r^{n_r},\end{array}$$ (79)

which is different from that in [6].

Remark 26 —

By Corollaries 8 and 9, the multivariable polynomials H _n1,…,nr ^(α)(x ₁,…, x _r) satisfy

$$\begin{array}{c}i{x}_i\frac{\partial }{\partial x_i}{H}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}=\left(n_i-i+\mathrm{1}\right)H_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right),\\ \sum _{i=\mathrm{1}}^{r}i{x}_i\frac{\partial }{\partial x_i}{H}_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}=\sum _{i=\mathrm{1}}^r\left(n_i-i+\mathrm{1}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times H_{n_{\mathrm{1}},\dots ,n_{i-\mathrm{1}},n_i-i+\mathrm{1},n_{i+\mathrm{1}},\dots ,n_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right).\end{array}$$ (80)

for n _i ≥ i − 1,  i = 1,2,…, r; n _j ≥ 0, j ≠ i.

Remark 27 —

From Theorem 11, the multivariable polynomials H _n1,…,nr ^(α)(x ₁,…, x _r) have the following addition formula:

$$\begin{array}{c}{H}_{n_{\mathrm{1}},\dots ,n_r}^{\left(\alpha +\beta \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}=\sum _{k_{\mathrm{1}}=\mathrm{0}}^{n_{\mathrm{1}}}\cdots \sum _{k_r=\mathrm{0}}^{n_r}{H}_{n_{\mathrm{1}}-k_{\mathrm{1}},\dots ,n_r-k_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times H_{k_{\mathrm{1}},\dots ,k_r}^{\left(\beta \right)}\left(x_{\mathrm{1}},\dots ,x_r\right).\end{array}$$ (81)

Furthermore, setting m _i = i, x _i = 0, y _i = −x _i, i = 1,2,…, r in Corollary 5, we obtain the following class of bilinear generating functions for the polynomials H _n1,…,nr ^(α)(x ₁,…, x _r).

Remark 28 —

If

$$\begin{array}{c}{\mathrm{\Xi }}_{\lambda ,\eta ,\alpha ,\beta ,\gamma }^{\mathbf{n},\mathbf{p}}\left(x_{\mathrm{1}},\dots ,x_r;u_{\mathrm{1}},\dots ,u_r;\mathbf{w}\right)\\ \hspace{1em}:=\sum _{k_{\mathrm{1}}=\mathrm{0}}^{\left[n_{\mathrm{1}}/p_{\mathrm{1}}\right]}\cdots \sum _{k_r=\mathrm{0}}^{\left[n_r/p_r\right]}{a}_{k_{\mathrm{1}},\dots ,k_r}{H}_{n_{\mathrm{1}}-p_{\mathrm{1}}{k}_{\mathrm{1}},\dots ,n_r-p_r{k}_r}^{\left(\alpha +\beta \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \times H_{{\lambda }_{\mathrm{1}}+{\eta }_{\mathrm{1}}{k}_{\mathrm{1}},\dots ,{\lambda }_r+{\eta }_r{k}_r}^{\left(\gamma \right)}\left(u_{\mathrm{1}},\dots ,u_r\right)w_{\mathrm{1}}^{k_{\mathrm{1}}}\cdots w_r^{k_r},\end{array}$$ (82)

where a _k1,…,kr ≠ 0; p _i ∈ ℕ; n _i, λ _i, η _i ∈ ℕ ₀, i = 1,2,…, r, and λ = (λ ₁,…, λ _r), η = (η ₁,…, η _r),w = (w ₁,…, w _r), then we have

$$\begin{array}{c}\sum _{l_{\mathrm{1}}=\mathrm{0}}^{n_{\mathrm{1}}}\cdots \sum _{l_r=\mathrm{0}}^{n_r}\hspace{0.17em}\sum _{k_{\mathrm{1}}=\mathrm{0}}^{\left[l_{\mathrm{1}}/p_{\mathrm{1}}\right]}\cdots \sum _{k_r=\mathrm{0}}^{\left[l_r/p_r\right]}{a}_{k_{\mathrm{1}},\dots ,k_r}{H}_{n_{\mathrm{1}}-l_{\mathrm{1}},\dots ,n_r-l_r}^{\left(\alpha \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times H_{l_{\mathrm{1}}-p_{\mathrm{1}}{k}_{\mathrm{1}},\dots ,l_r-p_r{k}_r}^{\left(\beta \right)}\left(x_{\mathrm{1}},\dots ,x_r\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times H_{{\lambda }_{\mathrm{1}}+{\eta }_{\mathrm{1}}{k}_{\mathrm{1}},\dots ,{\lambda }_r+{\eta }_r{k}_r}^{\left(\gamma \right)}\left(u_{\mathrm{1}},\dots ,u_r\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times w_{\mathrm{1}}^{k_{\mathrm{1}}}\cdots w_r^{k_r}\\ \hspace{1em}={\mathrm{\Xi }}_{\lambda ,\eta ,\alpha ,\beta ,\gamma }^{\mathbf{n},\mathbf{p}}\left(x_{\mathrm{1}},\dots ,x_r;u_{\mathrm{1}},\dots ,u_r;\mathbf{w}\right).\end{array}$$ (83)