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] 𝐶−𝑚𝑥𝑡+𝑦𝑡^𝑚𝑝=^∞_𝑛=0𝑃_𝑛𝑚,𝑥,𝑦,𝑝,𝐶𝑡^𝑛, (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 1−^𝑟_𝑖=1𝑚_𝑖𝑥_𝑖𝑡_𝑖−𝑦_𝑖𝑡^𝑚𝑖_𝑖^−𝛼=^∞_{𝑛1,…,𝑛𝑟=0}Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝐱,𝐲,𝐦)𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟,_𝑟𝑖=1𝑚_𝑖𝑥_𝑖t_𝑖−𝑦_𝑖𝑡^𝑚𝑖_𝑖<1, (2) where 𝐱=(𝑥₁,…,𝑥_𝑟),  𝐲=(𝑦₁,…,𝑦_𝑟), 𝐦=(𝑚₁,…,𝑚_𝑟) and 𝑚_𝑖(𝑖=1,2,…,𝑟) is a positive integer. It follows from (2) that Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}()=𝐱,𝐲,𝐦_{[𝑛1/𝑚1]𝑘1=0}⋯^{[𝑛𝑟/𝑚𝑟]}_𝑘𝑟=0(𝛼)_{𝑛1+⋯+𝑛𝑟−(𝑚1−1)𝑘1−⋯−(𝑚𝑟−1)𝑘𝑟}𝑘₁!⋯𝑘_𝑟!𝑛₁−𝑚₁𝑘₁!⋯𝑛_𝑟−𝑚_𝑟𝑘_𝑟!×(−1)^{𝑘1+⋯+𝑘𝑟}𝑚₁𝑥₁^{𝑛1−𝑚1𝑘1}⋯𝑚_𝑟𝑥_𝑟^{𝑛𝑟−𝑚𝑟𝑘𝑟}𝑦^𝑘1₁⋯𝑦^𝑘𝑟_𝑟, (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 𝑧₁,…,𝑧_𝑠(𝑠∈ℕ) and of complex order 𝜇, let Λ_𝜇,𝜈(𝐳;𝑤)∶=^∞_𝑘=0𝑎_𝑘Ω_{𝜇+𝜈𝑘}(𝐳)𝑤^𝑘, (4) where (𝑎_𝑘≠0,𝜇,𝜈∈ℂ); 𝐳=(𝑧₁,…,𝑧_𝑠). Then, for 𝑝∈ℕ; 𝐱=(𝑥₁,…,𝑥_𝑟); 𝐲=(𝑦₁,…,𝑦_𝑟); 𝐦=(𝑚₁,…,𝑚_𝑟), 𝑚_𝑖∈ℕ(𝑖=1,2,…,𝑟), one has ^∞_{𝑛1,…,𝑛𝑟=0}^[𝑛1/𝑝]_𝑘=0𝑎_𝑘Θ_𝑛^(𝛼)_{1−𝑝𝑘,𝑛2,…,𝑛𝑟}(𝐱,𝐲,𝐦)×𝑡^𝑛2₂⋯𝑡^𝑛𝑟_𝑟Ω_{𝜇+𝜈𝑘}(𝐳)𝜂^𝑘𝑡^𝑛1₁^−𝑝𝑘=1−^𝑟_𝑖=1𝑚_𝑖𝑥_𝑖𝑡_𝑖−𝑦_𝑖𝑡^𝑚𝑖_𝑖^−𝛼Λ_𝜇,𝜈𝐳;𝜂 (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 𝑛₁ by 𝑛₁+𝑝𝑘, we may write that 𝑆=^∞_{𝑛1,…,𝑛𝑟}^∞_=0𝑘=0𝑎_𝑘Θ_𝑛^(𝛼)_{1,𝑛2,…,𝑛𝑟}(𝐱,𝐲,𝐦)Ω_{𝜇+𝜈𝑘}(𝐳)𝜂^𝑘𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟=^∞_{𝑛1,…,𝑛𝑟=0}Θ_𝑛^(𝛼)_{1,𝑛2,…,𝑛𝑟}(𝐱,𝐲,𝐦)𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟^∞_𝑘=0𝑎_𝑘Ω_{𝜇+𝜈𝑘}(𝐳)𝜂^𝑘=1−^𝑟_𝑖=1𝑚_𝑖𝑥_𝑖𝑡_𝑖−𝑦_𝑖𝑡^𝑚𝑖_𝑖^−𝛼Λ_𝜇,𝜈𝐳;𝜂, (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 𝑧₁,…,𝑧_𝑠  (𝑠∈ℕ) and for 𝑝_𝑖∈ℕ, 𝜆_𝑖,𝜂_𝑖∈ℂ, 𝐳=(𝑧₁,…,𝑧_𝑠),𝐰=(𝑤₁,…,𝑤_𝑟), 𝜆∶=(𝜆₁,…,𝜆_𝑟),  𝜂∶=(𝜂₁,…,𝜂_𝑟) let Ω^𝐧,𝐩_{𝜆,𝜂,𝛼,𝛽,𝐦}(𝐱,𝐲;𝐳;𝐰)∶=^{[𝑛1/𝑝1]}_𝑘1=0⋯^{[𝑛𝑟/𝑝𝑟]}_𝑘𝑟=0𝑎_{𝑘1,…,𝑘𝑟}Θ_𝑛^(𝛼+𝛽)_{1−𝑝1𝑘1,…,𝑛𝑟−𝑝𝑟𝑘𝑟}(𝐱,𝐲,𝐦)×𝜑_{𝜆1+𝜂1𝑘1,…,𝜆𝑟+𝜂𝑟𝑘𝑟}(𝐳)𝑤^𝑘1₁⋯𝑤^𝑘𝑟_𝑟, (7) where 𝑎_{𝑘1,…,𝑘𝑟}≠0; 𝑛_𝑖,(𝑖=1,2,…,𝑟)∈ℕ₀; ℕ₀∶=ℕ∪{0}. Then, one has ^𝑛1_𝑙1=0⋯^𝑛𝑟_𝑙𝑟=0^{[𝑙1/𝑝1]}_𝑘1=0⋯^{[𝑙𝑟/𝑝𝑟]}_𝑘𝑟=0𝑎_{𝑘1,…,𝑘𝑟}Θ_𝑛^(𝛼)_{1−𝑙1,…,𝑛𝑟−𝑙𝑟}(𝐱,𝐲,𝐦)×Θ_𝑙^(𝛽)_{1−𝑝1𝑘1,…,𝑙𝑟−𝑝𝑟𝑘𝑟}(𝐱,𝐲,𝐦)×𝜑_{𝜆1+𝜂1𝑘1,…,𝜆𝑟+𝜂𝑟𝑘𝑟}(𝐳)×𝑤^𝑘1₁⋯𝑤^𝑘𝑟_𝑟=Ω^𝐧,𝐩_{𝜆,𝜂,𝛼,𝛽,𝐦}(𝐱,𝐲;𝐳;𝐰) (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=(𝑧₁,…,𝑧_𝑠),  𝑘∈ℕ₀, 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 𝑠=𝑟,Ω_{𝜇+𝜈𝑘}(𝐳)=g^{(𝛽1,…,𝛽𝑟)}_{𝜇+𝜈𝑘}(𝐳) (9) in Theorem 1, where the Chan-Chyan-Srivastava polynomials g^{(𝛽1,…,𝛽𝑟)}_{𝜇+𝜈𝑘}(𝐳) [10] are generated by ^𝑟_𝑖=11−𝑥_𝑖𝑡^−𝛼𝑖=^∞_𝑛=0g^{(𝛼1,…,𝛼𝑟)}_𝑛𝑥₁,…,𝑥_𝑟𝑡^𝑛𝛼_𝑖∈ℂ(𝑖=1,2,…,𝑟);|𝑡|<min𝑥₁⁻¹,…,𝑥_𝑟⁻¹. (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=0𝑎_𝑘g^{(𝛽1,…,𝛽𝑟)}_{𝜇+𝜈𝑘}(𝐳)𝑤^𝑘, 𝑎_𝑘≠0, 𝜇,𝜈∈ℂ, 𝐳=(𝑧₁,…,𝑧_𝑟) then, one has ^∞_{𝑛1,…,𝑛𝑟=0}^[𝑛1/𝑝]_𝑘=0𝑎_𝑘Θ_𝑛^(𝛼)_{1−𝑝𝑘,𝑛2,…,𝑛𝑟}(𝐱,𝐲,𝐦)×𝑡^𝑛2₂⋯𝑡^𝑛𝑟_𝑟g^{(𝛽1,…,𝛽𝑟)}_{𝜇+𝜈𝑘}(𝐳)𝜂^𝑘𝑡^𝑛1₁^−𝑝𝑘=1−^𝑟_𝑖=1𝑚_𝑖𝑥_𝑖𝑡_𝑖−𝑦_𝑖𝑡^𝑚𝑖_𝑖^−𝛼Λ_𝜇,𝜈𝐳;𝜂 (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 ^∞_{𝑛1,…,𝑛𝑟=0}^[𝑛1/𝑝]_𝑘=0Θ_𝑛^(𝛼)_{1−𝑝𝑘,𝑛2,…,𝑛𝑟}(𝐱,𝐲,𝐦)𝑡^𝑛2₂⋯𝑡^𝑛𝑟_𝑟g^{(𝛽1,…,𝛽𝑟)}_𝑘(𝐳)×𝜂^𝑘𝑡^𝑛1₁^−𝑝𝑘=1−^𝑟_𝑖=1𝑚_𝑖𝑥_𝑖𝑡_𝑖−𝑦_𝑖𝑡^𝑚𝑖_𝑖^𝑟−𝛼_𝑖=11−𝑧_𝑖𝜂^−𝛽𝑖,𝜂<min𝑧₁⁻¹,…,𝑧_𝑟⁻¹,^𝑟_𝑖=1𝑚_𝑖𝑥_𝑖𝑡_𝑖−𝑦_𝑖𝑡^𝑚𝑖_𝑖<1. (12)

On the other hand, choosing 𝑠=2𝑟and 𝜑_{𝜆1+𝜂1𝑘1,…,𝜆𝑟+𝜂𝑟𝑘𝑟}(𝐳)=Θ_𝜆^(𝛾)_{1+𝜂1𝑘1,…,𝜆𝑟+𝜂𝑟𝑘𝑟}(𝐮,𝐯,𝐦), (13)𝜆_𝑖,𝜂_𝑖(𝑖=1,2,…,𝑟)∈ℕ₀,  𝐮=(𝑢₁,…,𝑢_𝑟), 𝐯=(𝑣₁,…,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 Λ^𝐧,𝐩_{𝜆,𝜂,𝛼,𝛽,𝛾,𝐦}(𝐱,𝐲;𝐮,𝐯;𝐰)∶=^{[𝑛1/𝑝1]}_𝑘1=0⋯^{[𝑛𝑟/𝑝𝑟]}_𝑘𝑟=0𝑎_{𝑘1,…,𝑘𝑟}Θ_𝑛^(𝛼+𝛽)_{1−𝑝1𝑘1,…,𝑛𝑟−𝑝𝑟𝑘𝑟}(𝐱,𝐲,𝐦)×Θ_𝜆^(𝛾)_{1+𝜂1𝑘1,…,𝜆𝑟+𝜂𝑟𝑘𝑟}(𝐮,𝐯,𝐦)𝑤^𝑘1₁⋯𝑤^𝑘𝑟_𝑟, (14) where 𝑎_{𝑘1,…,𝑘𝑟}≠0; 𝑝_𝑖∈ℕ; 𝑛_𝑖,𝜆_𝑖,𝜂_𝑖∈ℕ₀, 𝑖=1,2,…,𝑟, then, we get ^𝑛1_𝑙1=0⋯^𝑛𝑟_𝑙𝑟=0^{[𝑙1/𝑝1]}_𝑘1=0⋯^{[𝑙𝑟/𝑝𝑟]}_𝑘𝑟=0𝑎_{𝑘1,…,𝑘𝑟}Θ_𝑛^(𝛼)_{1−𝑙1,…,𝑛𝑟−𝑙𝑟}(𝐱,𝐲,𝐦)×Θ_𝑙^(𝛽)_{1−𝑝1𝑘1,…,𝑙𝑟−𝑝𝑟𝑘𝑟}(𝐱,𝐲,𝐦)×Θ_𝜆^(𝛾)_{1+𝜂1𝑘1,…,𝜆𝑟+𝜂𝑟𝑘𝑟}(𝐮,𝐯,𝐦)×𝑤^𝑘1₁⋯𝑤^𝑘𝑟_𝑟=Λ^𝐧,𝐩_{𝜆,𝜂,𝛼,𝛽,𝛾,𝐦}(𝐱,𝐲;𝐮,𝐯;𝐰) (15) provided that each member of (15) exists.

Furthermore, for every suitable choice of the coefficients 𝑎_𝑘(𝑘∈ℕ₀), if the multivariable functions Ω_{𝜇+𝜈𝑘}(𝐲) and 𝜑_{𝜆1+𝜂1𝑘1,…,𝜆𝑟+𝜂𝑟𝑘𝑟}(𝐲),  𝐲=(𝑦₁,…,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 𝐺𝑚₁𝑥₁𝑡₁−𝑦₁𝑡^𝑚1₁,…,𝑚_𝑟𝑥_𝑟𝑡_𝑟−𝑦_𝑟𝑡^𝑚𝑟_𝑟=^∞_{𝑛1,…,𝑛𝑟=0}Ψ_{𝑛1,…,𝑛𝑟}(𝐱,𝐲,𝐦)𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟. (16) The following relations hold: 𝑛_𝑖Ψ_{𝑛1,…,𝑛𝑟}(𝐱,𝐲,𝐦)=𝑥_𝑖𝜕𝜕𝑥_𝑖Ψ_{𝑛1,…,𝑛𝑟}(𝐱,𝐲,𝐦)−𝑦_𝑖𝜕𝜕𝑥_𝑖Ψ_{𝑛1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖+1,𝑛𝑖+1,…,𝑛𝑟}(𝐱,𝐲,𝐦) (17) for 𝑛_𝑖≥𝑚_𝑖−1,  𝑖=1,2,…,𝑟; 𝑛_𝑗≥0, 𝑗≠𝑖 and 𝑛_𝑖Ψ_{𝑛1,…,𝑛𝑟}(𝐱,𝐲,𝐦)=𝑥_𝑖𝜕𝜕𝑥_𝑖Ψ_{𝑛1,…,𝑛𝑟}(𝐱,𝐲,𝐦) (18) for 𝑛_𝑖=0,1,…,𝑚_𝑖−2, 𝑖=1,2,…,𝑟; 𝑛_𝑗≥0, 𝑗≠𝑖. Also, one finds that 𝑚_𝑖𝑥_𝑖𝜕𝜕𝑦_𝑖Ψ_{𝑛1,…,𝑛𝑟}(𝐱,𝐲,𝐦)−𝑚_𝑖𝑦_𝑖𝜕𝜕𝑦_𝑖×Ψ_{𝑛1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖+1,𝑛𝑖+1,…,𝑛𝑟}()𝑛𝐱,𝐲,𝐦=−_𝑖−𝑚_𝑖+1Ψ_{𝑛1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖+1,𝑛𝑖+1,…,𝑛𝑟}(𝐱,𝐲,𝐦) (19) for 𝑛_𝑖≥𝑚_𝑖−1, 𝑖=1,2,…,𝑟; 𝑛_𝑗≥0, 𝑗≠𝑖 and 𝜕𝜕𝑦_𝑖Ψ_{𝑛1,…,𝑛𝑟}(𝐱,𝐲,𝐦)=0 (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 ^∞_{𝑛1,…,𝑛𝑟=0}𝑛_𝑖Ψ_{𝑛1,…,𝑛𝑟}(𝐱,𝐲,𝐦)𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟=𝑥_𝑖−𝑦_𝑖𝑡^𝑚𝑖_𝑖^−1∞_{𝑛1,…,𝑛𝑟=0}𝜕𝜕𝑥_𝑖Ψ_{𝑛1,…,𝑛𝑟}(𝐱,𝐲,𝐦)𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟=𝑥_𝑖^∞_{𝑛1,…,𝑛𝑟=0}𝜕𝜕𝑥_𝑖Ψ_{𝑛1,…,𝑛𝑟}(𝐱,𝐲,𝐦)𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟−𝑦_𝑖^∞_{𝑛1,…,𝑛𝑖−1,𝑛𝑖+1,…,𝑛𝑟𝑛=0𝑖≥𝑚𝑖−1}𝜕𝜕𝑥_𝑖Ψ_{𝑛1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖+1,𝑛𝑖+1,…,𝑛𝑟}(𝐱,𝐲,𝐦)×𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟. (21) Comparing the coefficients of 𝑡^𝑛1₁⋯𝑡^𝑛𝑟_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 −𝑚_𝑖𝜕𝜕𝑦_𝑖Ψ_{𝑛1,…,𝑛𝑖−1,𝑛𝑖−1,𝑛𝑖+1,…,𝑛𝑟}()=𝜕𝐱,𝐲,𝐦𝜕𝑥_𝑖Ψ_{𝑛1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖,𝑛𝑖+1,…,𝑛𝑟}(𝐱,𝐲,𝐦) (22) for 𝑛_𝑖≥𝑚_𝑖, 𝑖=1,2,…,𝑟; 𝑛_𝑗≥0, 𝑗≠𝑖 and 𝜕𝜕𝑦_𝑖Ψ_{𝑛1,…,𝑛𝑖−1,𝑛𝑖−1,𝑛𝑖+1,…,𝑛𝑟}(𝐱,𝐲,𝐦)=0 (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 −𝑚_𝑖𝑡_𝑖^∞_{𝑛1,…,𝑛𝑟=0}𝜕𝜕𝑦_𝑖Ψ_{𝑛1,…,𝑛𝑟}(𝐱,𝐲,𝐦)𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟=𝑡^𝑚𝑖_𝑖^∞_{𝑛1,…,𝑛𝑟=0}𝜕𝜕𝑥_𝑖Ψ_{𝑛1,…,𝑛𝑟}(𝐱,𝐲,𝐦)𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟, (24) which implies that −𝑚_𝑖𝜕𝜕𝑦_𝑖Ψ_{𝑛1,…,𝑛𝑖−1,𝑛𝑖−1,𝑛𝑖+1,…,𝑛𝑟}()=𝜕𝐱,𝐲,𝐦𝜕𝑥_𝑖Ψ_{𝑛1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖,𝑛𝑖+1,…,𝑛𝑟}(𝐱,𝐲,𝐦) (25) for 𝑛_𝑖≥𝑚_𝑖, 𝑖=1,2,…,𝑟; 𝑛_𝑗≥0, 𝑗≠𝑖 and 𝜕𝜕𝑦_𝑖Ψ_{𝑛1,…,𝑛𝑖−1,𝑛𝑖−1,𝑛𝑖+1,…,𝑛𝑟}(𝐱,𝐲,𝐦)=0 (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 𝐺𝑚₁𝑥₁𝑡₁−𝑦₁𝑡^𝑚1₁,…,𝑚_𝑟𝑥_𝑟𝑡_𝑟−𝑦_𝑟𝑡^𝑚𝑟_𝑟=1−^𝑟_𝑖=1𝑚_𝑖𝑥_𝑖𝑡_𝑖−𝑦_𝑖𝑡^𝑚𝑖_𝑖^−𝛼, (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 𝑛_𝑖Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝐱,𝐲,𝐦)=𝑥_𝑖𝜕𝜕𝑥_𝑖Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝐱,𝐲,𝐦)−𝑦_𝑖𝜕𝜕𝑥_𝑖Θ_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖+1,𝑛𝑖+1,…,𝑛𝑟}(),𝑚𝐱,𝐲,𝐦_𝑖𝑥_𝑖𝜕𝜕𝑦_𝑖Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝐱,𝐲,𝐦)−𝑚_𝑖𝑦_𝑖𝜕𝜕𝑦_𝑖×Θ_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖+1,𝑛𝑖+1,…,𝑛𝑟}()𝑛𝐱,𝐲,𝐦=−_𝑖−𝑚_𝑖+1Θ_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖+1,𝑛𝑖+1,…,𝑛𝑟}(𝐱,𝐲,𝐦) (28)(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 𝑛_𝑖Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝐱,𝐲,𝐦)=𝑥_𝑖𝜕𝜕𝑥_𝑖Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}(),𝜕𝐱,𝐲,𝐦𝜕𝑦_𝑖Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝐱,𝐲,𝐦)=0. (30)(31)

Corollary 9. By summing the relations given by (28) and (29), respectively, for 𝑖=1,2,…,𝑟, we get ^𝑟_𝑖=1𝑛_𝑖Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}()=𝐱,𝐲,𝐦_𝑟𝑖=1𝑥_𝑖𝜕𝜕𝑥_𝑖Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}()−𝐱,𝐲,𝐦_𝑟𝑖=1𝑦_𝑖𝜕𝜕𝑥_𝑖Θ_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖+1,𝑛𝑖+1,…,𝑛𝑟}(𝐱,𝐲,𝐦_𝑟𝑖=1),𝑚_𝑖𝑥_𝑖𝜕𝜕𝑦_𝑖Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}()−𝐱,𝐲,𝐦_𝑟𝑖=1𝑚_𝑖𝑦_𝑖𝜕𝜕𝑦_𝑖Θ_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖+1,𝑛𝑖+1,…,𝑛𝑟}(𝐱,𝐲,𝐦)=−^𝑟_𝑖=1𝑛_𝑖−𝑚_𝑖+1×Θ_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖+1,𝑛𝑖+1,…,𝑛𝑟}(𝐱,𝐲,𝐦) (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_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝐱,𝐲,𝐦) are −𝑚_𝑖𝜕𝜕𝑦_𝑖Θ_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−1,𝑛𝑖+1,…,𝑛𝑟}()=𝜕𝐱,𝐲,𝐦𝜕𝑥_𝑖Θ_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖,𝑛𝑖+1,…,𝑛𝑟}(𝐱,𝐲,𝐦) (33) for 𝑛_𝑖≥𝑚_𝑖, 𝑖=1,2,…,𝑟; 𝑛_𝑗≥0, 𝑗≠𝑖 and 𝜕𝜕𝑦_𝑖Θ_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−1,𝑛𝑖+1,…,𝑛𝑟}(𝐱,𝐲,𝐦)=0 (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) Θ_𝑛^(𝛼+𝛽)_{1,…,𝑛𝑟}(𝐱,𝐲,𝐦)=^𝑛1_𝑘1=0⋯^𝑛𝑟_𝑘𝑟=0Θ_𝑛^(𝛼)_{1−𝑘1,…,𝑛𝑟−𝑘𝑟}(𝐱,𝐲,𝐦)×Θ_𝑘^(𝛽)_{1,…,𝑘𝑟}(𝐱,𝐲,𝐦). (35)

Proof. From (2), we have ^∞_{𝑛1,…,𝑛𝑟=0}Θ_𝑛^(𝛼+𝛽)_{1,…,𝑛𝑟}(𝐱,𝐲,𝐦)𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟=1−^𝑟_𝑖=1𝑚_𝑖𝑥_𝑖𝑡_𝑖−𝑦_𝑖𝑡^𝑚𝑖_𝑖^{−(𝛼+𝛽)}=1−^𝑟_𝑖=1𝑚_𝑖𝑥_𝑖𝑡_𝑖−𝑦_𝑖𝑡^𝑚𝑖_𝑖^−𝛼×1−^𝑟_𝑖=1𝑚_𝑖𝑥_𝑖𝑡_𝑖−𝑦_𝑖𝑡^𝑚𝑖_𝑖^−𝛽=^∞_{𝑛1,…,𝑛𝑟=0}Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝐱,𝐲,𝐦)𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟×^∞_{𝑘1,…,𝑘𝑟=0}Θ_𝑘^(𝛽)_{1,…,𝑘𝑟}(𝐱,𝐲,𝐦)𝑡^𝑘1₁⋯𝑡^𝑘𝑟_𝑟. (36) Replacing 𝑛_𝑖 by 𝑛_𝑖−𝑘_𝑖, 𝑖=1,2,…,𝑟,   the right-hand side of the last equality is ^∞_{𝑛1,…,𝑛𝑟}^𝑛₌₀¹_𝑘1=0⋯^𝑛𝑟_𝑘𝑟=0Θ_𝑛^(𝛼)_{1−𝑘1,…,𝑛𝑟−𝑘𝑟}(𝐱,𝐲,𝐦)×Θ_𝑘^(𝛽)_{1,…,𝑘𝑟}(𝐱,𝐲,𝐦)𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟. (37) Comparing the coefficients of 𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟 completes the proof.

We now give expansions of the family of multivariable polynomials Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝐱,𝐲,𝐦) given explicitly by (3) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials.

Theorem 12. Expansions of Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝐱,𝐲,𝐦) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are as follows Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}()=𝐱,𝐲,𝐦_{𝑟𝑖=1[𝑛𝑖/𝑚𝑖]𝑘𝑖=0[(𝑛𝑖−𝑚𝑖𝑘𝑖)/2]𝑠𝑖=0}2𝑛_𝑖−2𝑚_𝑖𝑘_𝑖−4𝑠_𝑖+1𝑘_𝑖!𝑠_𝑖!(3/2)_{𝑛𝑖−𝑚𝑖𝑘𝑖−𝑠𝑖}×(−1)^𝑘𝑖𝑦^𝑘𝑖_𝑖𝑃_{𝑛𝑖−𝑚𝑖𝑘𝑖−2𝑠𝑖}𝑚_𝑖𝑥_𝑖2×(𝛼)_{𝑛1+⋯+𝑛𝑟−(𝑚1−1)𝑘1−⋯−(𝑚𝑟−1)𝑘𝑟},Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}()=𝐱,𝐲,𝐦_{𝑟𝑖=1[𝑛𝑖/𝑚𝑖]𝑘𝑖=0[(𝑛𝑖−𝑚𝑖𝑘𝑖)/2]𝑠𝑖=0}𝜈_𝑖+𝑛_𝑖−𝑚_𝑖𝑘_𝑖−2𝑠_𝑖𝑘_𝑖!𝑠_𝑖!𝜈_{𝑖𝑛𝑖−𝑚𝑖𝑘𝑖−𝑠𝑖+1}×−𝑦_𝑖^𝑘𝑖𝐶^𝜈𝑖_{𝑛𝑖−𝑚𝑖𝑘𝑖−2𝑠𝑖}𝑚_𝑖𝑥_𝑖2×(𝛼)_{𝑛1+⋯+𝑛𝑟−(𝑚1−1)𝑘1−⋯−(𝑚𝑟−1)𝑘𝑟},Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}()=𝐱,𝐲,𝐦_{𝑟𝑖=1[𝑛𝑖/𝑚𝑖]𝑘𝑖=0[(𝑛𝑖−𝑚𝑖𝑘𝑖)/2]𝑠𝑖=0}−𝑦_𝑖^𝑘𝑖𝐻_{𝑛𝑖−𝑚𝑖𝑘𝑖−2𝑠𝑖}𝑚_𝑖𝑥_𝑖/2𝑘_𝑖!𝑠_𝑖!𝑛_𝑖−𝑚_𝑖𝑘_𝑖−2𝑠_𝑖!×(𝛼)_{𝑛1+⋯+𝑛𝑟−(𝑚1−1)𝑘1−⋯−(𝑚𝑟−1)𝑘𝑟},Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}()=𝐱,𝐲,𝐦_{𝑟𝑖=1[𝑛𝑖/𝑚𝑖]𝑘𝑖𝑛=0𝑖−𝑚𝑖𝑘𝑖𝑠𝑖=0}𝛽_𝑖+1_{𝑛𝑖−𝑚𝑖𝑘𝑖}2^{𝑛𝑖−𝑚𝑖𝑘𝑖}(−1)^𝑠𝑖𝑘_𝑖!𝑛_𝑖−𝑚_𝑖𝑘_𝑖−𝑠_𝑖!𝛽_𝑖+1_𝑠𝑖×−𝑦_𝑖^𝑘𝑖𝐿^(𝛽𝑖)_𝑠𝑖𝑚_𝑖𝑥_𝑖2×(𝛼)_{𝑛1+⋯+𝑛𝑟−(𝑚1−1)𝑘1−⋯−(𝑚𝑟−1)𝑘𝑟}. (38)

Proof. By (2), we get ^∞_{𝑛1,…,𝑛𝑟=0}Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝐱,𝐲,𝐦)𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟=1−^𝑟_𝑖=1𝑚_𝑖𝑥_𝑖𝑡_𝑖−𝑦_𝑖𝑡^𝑚𝑖_𝑖^−𝛼=^∞_{𝑛1,…,𝑛𝑟}^∞_{=0𝑘1,…,𝑘𝑟=0}(𝛼)_{𝑛1+⋯+𝑛𝑟+𝑘1+⋯+𝑘𝑟}𝑘₁!⋯𝑘_𝑟!𝑛₁!⋯𝑛_𝑟!(−1)^{𝑘1+⋯+𝑘𝑟}×𝑚₁𝑥₁^𝑛1⋯𝑚_𝑟𝑥_𝑟^𝑛𝑟𝑦^𝑘1₁⋯𝑦^𝑘𝑟_𝑟×𝑡^{𝑛1+𝑚1𝑘1}₁⋯𝑡^{𝑛𝑟+𝑚𝑟𝑘𝑟}_𝑟. (39) If we use the result in [11, page 181] (𝑚𝑥)^𝑛=𝑛!^[𝑛/2]_𝑠=0(2𝑛−4𝑠+1)𝑃_𝑛−2𝑠(𝑚𝑥/2)𝑠!(3/2)_𝑛−𝑠, (40) we can write that ^∞_{𝑛1,…,𝑛𝑟=0}Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝐱,𝐲,𝐦)𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟=^𝑟_𝑖=1^∞_𝑛𝑖^∞_=0𝑘𝑖=0^[𝑛𝑖/2]_𝑠𝑖=02𝑛_𝑖−4𝑠_𝑖+1𝑠_𝑖!𝑘_𝑖!(3/2)_{𝑛𝑖−𝑠𝑖}𝑃_{𝑛𝑖−2𝑠𝑖}𝑚_𝑖𝑥_𝑖2×−𝑦_𝑖^𝑘𝑖𝑡^{𝑛𝑖+𝑚𝑖𝑘𝑖}_𝑖(𝛼)_{𝑛1+⋯+𝑛𝑟+𝑘1+⋯+𝑘𝑟}. (41) Getting 𝑛_𝑖−𝑚_𝑖𝑘_𝑖 instead of 𝑛_𝑖 in the last equality, we have ^∞_{𝑛1,…,𝑛𝑟=0}Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝐱,𝐲,𝐦)𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟=^∞_{𝑛1,…,𝑛𝑟}^𝑟_=0𝑖=1^{[𝑛𝑖/𝑚𝑖]}_𝑘𝑖=0^{[(𝑛𝑖−𝑚𝑖𝑘𝑖)/2]}_𝑠𝑖=02𝑛_𝑖−2𝑚_𝑖𝑘_𝑖−4𝑠_𝑖+1𝑠_𝑖!𝑘_𝑖!(3/2)_{𝑛𝑖−𝑚𝑖𝑘𝑖−𝑠𝑖}×𝑃_{𝑛𝑖−𝑚𝑖𝑘𝑖−2𝑠𝑖}𝑚_𝑖𝑥_𝑖2−𝑦_𝑖^𝑘𝑖𝑡^𝑛𝑖_𝑖×(𝛼)_{𝑛1+⋯+𝑛𝑟−(𝑚1−1)𝑘1−⋯−(𝑚𝑟−1)𝑘𝑟}=^∞_{𝑛1,…,𝑛𝑟=0}^𝑟_𝑖=1^{[𝑛𝑖/𝑚𝑖]}_𝑘𝑖=0^{[(𝑛𝑖−𝑚𝑖𝑘𝑖)/2]}_𝑠𝑖=02𝑛_𝑖−2𝑚_𝑖𝑘_𝑖−4𝑠_𝑖+1𝑠_𝑖!𝑘_𝑖!(3/2)_{𝑛𝑖−𝑚𝑖𝑘𝑖−𝑠𝑖}×𝑃_{𝑛𝑖−𝑚𝑖𝑘𝑖−2𝑠𝑖}𝑚_𝑖𝑥_𝑖2−𝑦_𝑖^𝑘𝑖×(𝛼)_{𝑛1+⋯+𝑛𝑟−(𝑚1−1)𝑘1−⋯−(𝑚𝑟−1)𝑘𝑟}𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟. (42) Comparing the coefficients of 𝑡^𝑛1₁⋯𝑡^𝑛𝑟_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)] (𝑚𝑥)^𝑛=𝑛!^[𝑛/2]_𝑘=0(𝜈+𝑛−2𝑘)𝐶^𝜈_𝑛−2𝑘(𝑚𝑥/2)𝑘!(𝜈)_{𝑛+1−𝑘},(𝑚𝑥)^𝑛=𝑛!^[𝑛/2]_𝑘=0𝐻_𝑛−2𝑘(𝑚𝑥/2)()!,()𝑘!𝑛−2𝑘𝑚𝑥^𝑛𝑛!=2^𝑛𝑛_𝑘=0(−1)^𝑘(𝛼+1)_𝑛𝐿_𝑘^(𝛼)(𝑚𝑥/2)(𝑛−𝑘)!(𝛼+1)_𝑘, (43) one can easily obtain the other expansions of Θ_𝑛^(𝛼)_{1,…,𝑛𝑟}(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 1−2𝑥₁𝑡₁+𝑡²₁−⋯−2𝑥_𝑟𝑡_𝑟+𝑡²_𝑟^−𝛼=^∞_{𝑛1,…,𝑛𝑟=0}𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟, (44) which reduces to two variables Gegenbauer polynomials given by [12] for 𝑟=2. Equation (44) yields the following formula: 𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^[𝑛1/2]_𝑘1=0⋯^[𝑛𝑟/2]_𝑘𝑟=0(𝛼)_{𝑛1+⋯+𝑛𝑟−𝑘1−⋯−𝑘𝑟}×𝑘₁!⋯𝑘_𝑟!𝑛₁−2𝑘₁!⋯𝑛_𝑟−2𝑘_𝑟!⁻¹×(−1)^{𝑘1+⋯+𝑘𝑟}2𝑥₁^{𝑛1−2𝑘1}⋯2𝑥_𝑟^{𝑛𝑟−2𝑘𝑟}. (45)

It follows that 𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟) is a polynomial of degree 𝑛_𝑖 with respect to the fixed variable 𝑥_𝑖(𝑖=1,2,…,𝑟).  Thus, 𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟) is a polynomial of total degree (𝑛₁+⋯+𝑛_𝑟) with respect to the variables 𝑥₁,…,𝑥_𝑟. Equation (45) also yields 𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=2^{𝑛1+⋯+𝑛𝑟}(𝛼)_{𝑛1+⋯+𝑛𝑟}𝑛₁!⋯𝑛_𝑟!𝑥^𝑛1₁⋯𝑥^𝑛𝑟_𝑟+Π𝑥₁,…,𝑥_𝑟, (46) where Π(𝑥₁,…,𝑥_𝑟) is a polynomial of degree (𝑛₁+⋯+𝑛_𝑟−2) with respect to the variables 𝑥₁,…,𝑥_𝑟. In (44), by getting 𝑥₁→−𝑥₁ and 𝑡₁→−𝑡₁, we have 𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}−𝑥₁,𝑥₂,…,𝑥_𝑟=(−1)^𝑛1𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟. (47)

Similarly, for 𝑖=1,2,…,𝑟,   we get 𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑖−1,−𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑟=(−1)^𝑛𝑖𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟. (48)

Taking 𝑥_𝑖→−𝑥_𝑖 and𝑡_𝑖→−𝑡_𝑖, 𝑖=1,2,…,𝑟, in (44), we obtain 𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}−𝑥₁,…,−𝑥_𝑟=(−1)^{𝑛1+⋯+𝑛𝑟}𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟. (49)

Theorem 13. For the polynomials 𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟), one has 𝐶^(𝛼)_{2𝑛1,…,2𝑛𝑟}()=()0,0,…,0−1^{𝑛1+⋯+𝑛𝑟}(𝛼)_{𝑛1+⋯+𝑛𝑟}𝑛₁!⋯𝑛_𝑟! (50) and if at least one of 𝑛_𝑖, 𝑖=1,2,…,𝑟, is odd; then 𝐶_𝑛^(𝛼)_{1,….,𝑛𝑟}(0,0,…,0)=0. (51)

Proof. If we set all 𝑥_𝑖=0, 𝑖=1,2,…,𝑟 in (44), we have 1+𝑡²₁+⋯+𝑡²_𝑟^−𝛼=^∞_{𝑛1,…,𝑛𝑟=0}𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}(0,0,…,0)𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟. (52) On the other hand, we get 1+𝑡²₁+⋯+𝑡²_𝑟^−𝛼=^∞_{𝑛1,…,𝑛𝑟=0}(−1)^{𝑛1+⋯+𝑛𝑟}𝑛₁!⋯𝑛_𝑟!(𝛼)_{𝑛1+⋯+𝑛𝑟}𝑡^2𝑛1₁⋯𝑡^2𝑛𝑟_𝑟. (53) By comparing the coefficients of 𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟, we obtain the desired.

From the theorems and corollaries given in Section 4, we can give some other properties of 𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟).

Remark 14. By Corollaries 8 and 9, for the family of multivariable polynomials generated by (44), the following relations: 𝑛_𝑖𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=𝑥_𝑖𝜕𝜕𝑥_𝑖𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟−𝜕𝜕𝑥_𝑖𝐶_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−1,𝑛𝑖+1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟,_𝑟𝑖=1𝑛_𝑖𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^𝑟_𝑖=1𝑥_𝑖𝜕𝜕𝑥_𝑖𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟−^𝑟_𝑖=1𝜕𝜕𝑥_𝑖𝐶_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−1,𝑛𝑖+1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟 (54) hold for 𝑛_𝑖≥1,𝑖=1,2,…,𝑟.

Remark 15. From Theorem 11, the multivariable polynomials 𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟) satisfy the following addition formula: 𝐶_𝑛^(𝛼+𝛽)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^𝑛1_𝑘1=0⋯^𝑛𝑟_𝑘𝑟=0𝐶_𝑛^(𝛼)_{1−𝑘1,…,𝑛𝑟−𝑘𝑟}𝑥₁,…,𝑥_𝑟×𝐶_𝑘^(𝛽)_{1,…,𝑘𝑟}𝑥₁,…,𝑥_𝑟. (55)

Remark 16. As a result of Theorem 12, expansions of 𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are as follows: 𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^𝑟_𝑖=1^[𝑛𝑖/2]_𝑘𝑖=0^{[(𝑛𝑖−2𝑘𝑖)/2]}_𝑠𝑖=02𝑛_𝑖−4𝑘_𝑖−4𝑠_𝑖+1𝑘_𝑖!𝑠_𝑖!(3/2)_{𝑛𝑖−2𝑘𝑖−𝑠𝑖}×(−1)^𝑘𝑖𝑃_{𝑛𝑖−2𝑘𝑖−2𝑠𝑖}𝑥_𝑖×(𝛼)_{𝑛1+⋯+𝑛𝑟−𝑘1−⋯−𝑘𝑟},𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^𝑟_𝑖=1^[𝑛𝑖/2]_𝑘𝑖=0^{[(𝑛𝑖−2𝑘𝑖)/2]}_𝑠𝑖=0𝜈_𝑖+𝑛_𝑖−2𝑘_𝑖−2𝑠_𝑖𝑘_𝑖!𝑠_𝑖!𝜈_{𝑖𝑛𝑖−2𝑘𝑖−𝑠𝑖+1}×(−1)^𝑘𝑖𝐶^𝜈𝑖_{𝑛𝑖−2𝑘𝑖−2𝑠𝑖}𝑥_𝑖×(𝛼)_{𝑛1+⋯+𝑛𝑟−𝑘1−⋯−𝑘𝑟},𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^𝑟_𝑖=1^[𝑛𝑖/2]_𝑘𝑖=0^{[(𝑛𝑖−2𝑘𝑖)/2]}_𝑠𝑖=0(−1)^𝑘𝑖𝐻_{𝑛𝑖−2𝑘𝑖−2𝑠𝑖}𝑥_𝑖𝑘_𝑖!𝑠_𝑖!𝑛_𝑖−2𝑘_𝑖−2𝑠_𝑖!×(𝛼)_{𝑛1+⋯+𝑛𝑟−𝑘1−⋯−𝑘𝑟,}𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^𝑟_𝑖=1^[𝑛𝑖/2]_𝑘𝑖^𝑛₌₀^{𝑖−2𝑘𝑖}_𝑠𝑖=0𝛽_𝑖+1_{𝑛𝑖−2𝑘𝑖}2^{𝑛𝑖−2𝑘𝑖}(−1)^𝑠𝑖𝑘_𝑖!𝑛_𝑖−2𝑘_𝑖−𝑠_𝑖!𝛽_𝑖+1_𝑠𝑖×(−1)^𝑘𝑖𝐿^(𝛽𝑖)_𝑠𝑖𝑥_𝑖×(𝛼)_{𝑛1+⋯+𝑛𝑟−𝑘1−⋯−𝑘𝑟}. (56)

We now give a hypergeometric representation for the multivariable polynomials 𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟) given by (44).

Theorem 17. The multivariable polynomials 𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟) have the following hypergeometric representation: 𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=(𝛼)_{𝑛1+⋯+𝑛𝑟}𝑛₁!⋯𝑛_𝑟!2𝑥₁^𝑛1⋯2𝑥_𝑟^𝑛𝑟×𝐹_𝐵^(𝑟)−𝑛₁2𝑛,…,−^𝑟2,−𝑛₁+12,…,−𝑛_𝑟+12;1−𝛼−𝑛₁−⋯−𝑛_𝑟;1𝑥²₁1,…,𝑥²_𝑟,1max𝑥²₁1,…,𝑥²_𝑟<1, (57) where 𝐹_𝐵^(𝑟) is a Lauricella function of 𝑟-variable defined by [13] (see also [2]) 𝐹_𝐵^(𝑟)𝑎₁,…,𝑎_𝑟,𝑏₁,…,𝑏_𝑟;𝑐;𝑥₁,…,𝑥_𝑟=^∞_{𝑛1,…,𝑛𝑟=0}𝑎_1𝑛1⋯𝑎_𝑟𝑛𝑟𝑏_1𝑛1⋯𝑏_𝑟𝑛𝑟(𝑐)_{𝑛1+⋯+𝑛𝑟}𝑥^𝑛1₁𝑛₁!⋯𝑥^𝑛𝑟_𝑟𝑛_𝑟!,max𝑥₁,…,𝑥_𝑟<1. (58)

Proof. We have the following results from [11]: (𝛼)_𝑛−𝑘=(−1)^𝑘(𝛼)_𝑛(1−𝛼−𝑛)_𝑘,(−𝑛)_2𝑘=2^2𝑘−𝑛2_𝑘−𝑛+12_𝑘,(𝑛−2𝑘)!=𝑛!(−𝑛)_2𝑘. (59) In view of these relations, the equality (45) can be written as follows: 𝐶_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^[𝑛1/2]_𝑘1=0⋯^[𝑛𝑟/2]_𝑘𝑟=0(𝛼)_{𝑛1+⋯+𝑛𝑟}2^{2(𝑘1+⋯+𝑘𝑟)}−𝑛₁2_𝑘1×−𝑛₁+12_𝑘1⋯−𝑛_𝑟2_𝑘𝑟×−𝑛_𝑟+12_𝑘𝑟×𝑘₁!⋯𝑘_𝑟!𝑛₁!⋯𝑛_𝑟!×1−𝛼−𝑛₁−⋯−𝑛_{𝑟𝑘1+⋯+𝑘𝑟}⁻¹×2𝑥₁^{𝑛1−2𝑘1}⋯2𝑥_𝑟^{𝑛𝑟−2𝑘𝑟}=(𝛼)_{𝑛1+⋯+𝑛𝑟}𝑛₁!⋯𝑛_𝑟!2𝑥₁^𝑛1⋯2𝑥_𝑟^𝑛𝑟×^[𝑛1/2]_𝑘1=0⋯^[𝑛𝑟/2]_𝑘𝑟=0−𝑛₁2_𝑘1−𝑛₁+12_𝑘1⋯−𝑛_𝑟2_𝑘𝑟−𝑛_𝑟+12_𝑘𝑟×1−𝛼−𝑛₁−⋯−𝑛_{𝑟𝑘1+⋯+𝑘𝑟}×𝑘₁!⋯𝑘_𝑟!⁻¹×1𝑥²₁^𝑘1⋯1𝑥²_𝑟^𝑘𝑟=𝐹_𝐵^(𝑟)−𝑛₁2𝑛,…,−^𝑟2,−𝑛₁+12,…,−𝑛_𝑟+12;1−𝛼−𝑛₁−⋯−𝑛_𝑟;1𝑥²₁1,…,𝑥²_𝑟×(𝛼)_{𝑛1+⋯+𝑛𝑟}𝑛₁!⋯𝑛_𝑟!2𝑥₁^𝑛1⋯2𝑥_𝑟^𝑛𝑟, (60) where max{1/𝑥²₁,…,1/𝑥²_𝑟}<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 1−𝑥₁𝑡^𝑚1₁−𝑥₂𝑡^𝑚2₂−⋯−𝑥_𝑟𝑡^𝑚𝑟_𝑟^−𝛼=^∞_{𝑛1,…,𝑛𝑟=0}𝑈_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟, (61) which is a different unification from that in [8].

Remark 19. From Corollaries 8 and 9, the multivariable polynomials 𝑈_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟) satisfy 𝑚_𝑖𝑥_𝑖𝜕𝜕𝑥_𝑖𝑈_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖+1,𝑛𝑖+1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=𝑛_𝑖−𝑚_𝑖+1𝑈_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖+1,𝑛𝑖+1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟,_𝑟𝑖=1𝑚_𝑖𝑥_𝑖𝜕𝜕𝑥_𝑖×𝑈_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖+1,𝑛𝑖+1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^𝑟_𝑖=1𝑛_𝑖−𝑚_𝑖+1×𝑈_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−𝑚𝑖+1,𝑛𝑖+1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟. (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_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟) have the following addition formula: 𝑈_𝑛^(𝛼+𝛽)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^𝑛1_𝑘1=0⋯^𝑛𝑟_𝑘𝑟=0𝑈_𝑛^(𝛼)_{1−𝑘1,…,𝑛𝑟−𝑘𝑟}𝑥₁,…,𝑥_𝑟×𝑈_𝑘^(𝛽)_{1,…,𝑘𝑟}𝑥₁,…,𝑥_𝑟. (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] 1−𝑥₁𝑡₁−⋯−𝑥_𝑟𝑡_𝑟^−𝛼=^∞_{𝑛1,…,𝑛𝑟=0}𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟. (64)

They are given explicitly by 𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=(𝛼)_{𝑛1+⋯+𝑛𝑟}𝑛₁!⋯𝑛_𝑟!𝑥^𝑛1₁⋯𝑥^𝑛𝑟_𝑟. (65)

Remark 21. From Corollaries 8 and 9, the multivariable polynomials 𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟) satisfy 𝑛_𝑖𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=𝑥_𝑖𝜕𝜕𝑥_𝑖𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟 (66) for 𝑖=1,2,…,𝑟 and ^𝑟_𝑖=1𝑛_𝑖𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^𝑟_𝑖=1𝑥_𝑖𝜕𝜕𝑥_𝑖𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟. (67)

Remark 22. By Theorem 11, the multivariable polynomials 𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟) have the following addition formula: 𝐺_𝑛^(𝛼+𝛽)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^𝑛1_𝑘1=0⋯^𝑛𝑟_𝑘𝑟=0𝐺_𝑛^(𝛼)_{1−𝑘1,…,𝑛𝑟−𝑘𝑟}𝑥₁,…,𝑥_𝑟×𝐺_𝑘^(𝛽)_{1,…,𝑘𝑟}𝑥₁,…,𝑥_𝑟. (68)

Remark 23. From Theorem 12, expansions of 𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are given by 𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^𝑟_𝑖=1^[𝑛𝑖/2]_𝑠𝑖=02𝑛_𝑖−4𝑠_𝑖+1𝑠_𝑖!(3/2)_{𝑛𝑖−𝑠𝑖}𝑃_{𝑛𝑖−2𝑠𝑖}𝑥_𝑖2×(𝛼)_{𝑛1+⋯+𝑛𝑟},𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^𝑟_𝑖=1^[𝑛𝑖/2]_𝑠𝑖=0𝜈_𝑖+𝑛_𝑖−2𝑠_𝑖𝑠_𝑖!𝜈_{𝑖𝑛𝑖−𝑠𝑖+1}𝐶^𝜈𝑖_{𝑛𝑖−2𝑠𝑖}𝑥_𝑖2×(𝛼)_{𝑛1+⋯+𝑛𝑟},𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^𝑟_𝑖=1^[𝑛𝑖/2]_𝑠𝑖=0𝐻_{𝑛𝑖−2𝑠𝑖}𝑥_𝑖/2𝑠_𝑖!𝑛_𝑖−2𝑠_𝑖!(𝛼)_{𝑛1+⋯+𝑛𝑟},𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^𝑟_𝑖=1^𝑛𝑖_𝑠𝑖=0𝛽_𝑖+1_𝑛𝑖2^𝑛𝑖(−1)^𝑠𝑖𝑛_𝑖−𝑠_𝑖!𝛽_𝑖+1_𝑠𝑖𝐿^(𝛽𝑖)_𝑠𝑖𝑥_𝑖2×(𝛼)_{𝑛1+⋯+𝑛𝑟}. (69)

We can discuss some generating functions for the multivariable polynomials 𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟).

Theorem 24. For the polynomials 𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟),   the following generating function holds true for 𝑘∈ℕ₀: ^∞_{𝑛1,…,𝑛𝑟=0}𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟𝑘+𝑛_𝑖𝑚𝑧^𝑛1₁⋯𝑧^𝑛𝑟_𝑟=𝑚!1−𝑥₁𝑧₁−⋯−𝑥_𝑟𝑧_𝑟^−𝛼×g_𝑚^{(𝛼,−𝑘)}𝑥_𝑖𝑧_𝑖1−𝑥₁𝑧₁−⋯−𝑥_𝑟𝑧_𝑟,−1, (70) for each 𝑖=1,2,…,𝑟, where [𝑘]_𝑚∶=𝑘(𝑘−1)⋯(𝑘−𝑚+1),𝑚∈ℕ,[𝑘]₀=1 (71) and g_𝑛^(𝛼,𝛽)(𝑥,𝑦) denotes the classical Lagrange polynomials defined by [14] (1−𝑥𝑡)^−𝛼1−𝑦𝑡^−𝛽=^∞_𝑛=0g_𝑛^(𝛼,𝛽)𝑥,𝑦𝑡^𝑛𝛼,𝛽∈ℂ;|𝑡|<min|𝑥|⁻¹,𝑦⁻¹. (72)

Proof. Fix 𝑖=1,2,…,𝑟. Let 𝒞_𝜉𝑖and 𝒞^∗_𝜉𝑖 denote the circles of radius 𝜀_𝑖, centered 𝜉_𝑖=𝑧_𝑖 and 𝜉_𝑖=0, respectively, where 0<𝜀_𝑖<𝑥_𝑖⁻¹1−^𝑟_𝑘=1𝑥_𝑘𝑧_𝑘. (73) Here these circles are described in the positive direction. By Cauchy's Integral Formula, we have 𝐷^𝑚_𝑧𝑖1−𝑥₁𝑧₁−⋯−𝑥_𝑟𝑧_𝑟^−𝛼𝑧^𝑘_𝑖=𝑚!2𝜋𝑖_𝒞𝜉𝑖𝜉^𝑘_𝑖1−𝑥₁𝑧₁−⋯−𝑥_𝑖𝜉_𝑖−⋯−𝑥_𝑟𝑧_𝑟^−𝛼𝜉_𝑖−𝑧_𝑖^𝑚+1𝑑𝜉_𝑖=𝑚!×2𝜋𝑖_{𝒞∗𝜉𝑖}𝜉_𝑖+𝑧_𝑖^𝑘1−𝑥₁𝑧₁−⋯−𝑥_𝑖𝜉_𝑖+𝑧_𝑖−⋯−𝑥_𝑟𝑧_𝑟^−𝛼𝜉_𝑖^𝑚+1𝑑𝜉_𝑖=𝑚!𝑧2𝜋𝑖^𝑘𝑖1−𝑥₁𝑧₁−⋯−𝑥_𝑟𝑧_𝑟^−𝛼×_{𝒞∗𝜉𝑖}1+𝜉_𝑖/𝑧_𝑖^𝑘𝜉_𝑖^𝑚+1𝑥1−^𝑖𝜉_𝑖1−𝑥₁𝑧₁−⋯−𝑥_𝑟𝑧_𝑟^−𝛼𝑑𝜉_𝑖=𝑚!𝑧^𝑘_𝑖1−𝑥₁𝑧₁−⋯−𝑥_𝑟𝑧_𝑟^−𝛼×g_𝑚^{(𝛼,−𝑘)}𝑥_𝑖1−𝑥₁𝑧₁−⋯−𝑥_𝑟𝑧_𝑟1,−𝑧_𝑖=𝑚!𝑧_𝑖^𝑘−𝑚1−𝑥₁𝑧₁−⋯−𝑥_𝑟𝑧_𝑟^−𝛼×g_𝑚^{(𝛼,−𝑘)}𝑥_𝑖𝑧_𝑖1−𝑥₁𝑧₁−⋯−𝑥_𝑟𝑧_𝑟,−1. (74) On the other hand, we can write that 𝐷^𝑚_𝑧𝑖1−𝑥₁𝑧₁−⋯−𝑥_𝑟𝑧_𝑟^−𝛼𝑧^𝑘_𝑖=𝐷^𝑚_𝑧𝑖𝑧^𝑘_𝑖^∞_{𝑛1,…,𝑛𝑟=0}𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟𝑧^𝑛1₁⋯𝑧^𝑛𝑟_𝑟=^∞_{𝑛1,…,𝑛𝑟=0}𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟𝑑^𝑚𝑑𝑧^𝑚_𝑖𝑧^𝑛1₁⋯𝑧^𝑛𝑖_𝑖^+𝑘⋯𝑧^𝑛𝑟_𝑟=𝑧_𝑖^{∞𝑘−𝑚}_{𝑛1,…,𝑛𝑟=0}𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟𝑘+𝑛_𝑖𝑚𝑧^𝑛1₁⋯𝑧^𝑛𝑟_𝑟. (75) From (74) and (75), we see that ^∞_{𝑛1,…,𝑛𝑟=0}𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟𝑘+𝑛_𝑖𝑚𝑧^𝑛1₁⋯𝑧^𝑛𝑟_𝑟=𝑚!1−𝑥₁𝑧₁−⋯−𝑥_𝑟𝑧_𝑟^−𝛼×g_𝑚^{(𝛼,−𝑘)}𝑥_𝑖𝑧_𝑖1−𝑥₁𝑧₁−⋯−𝑥_𝑟𝑧_𝑟,−1. (76)

Corollary 25. From (71), one observes that 𝑛_𝑖+𝑚_𝑚=𝑛_𝑖+𝑚𝑛_𝑖+𝑚−1⋯𝑛_𝑖=𝑛+1_𝑖+1_𝑚 (77) for each 𝑖=1,2,…,𝑟. By setting 𝑘=𝑚in (70) and then using (77), one has ^∞_{𝑛1,…,𝑛𝑟=0}𝑛_𝑖+1_𝑚𝐺_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟𝑧^𝑛1₁⋯𝑧^𝑛𝑟_𝑟=𝑚!1−𝑥₁𝑧₁−⋯−𝑥_𝑟𝑧_𝑟^−𝛼×g_𝑚^{(𝛼,−𝑚)}𝑥_𝑖𝑧_𝑖1−𝑥₁𝑧₁−⋯−𝑥_𝑟𝑧_𝑟,−1. (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 1−𝑥₁𝑡₁−𝑥₂𝑡²₂−⋯−𝑥_𝑟𝑡^𝑟_𝑟^−𝛼=^∞_{𝑛1,…,𝑛𝑟=0}𝐻_𝑛^(𝛼)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟𝑡^𝑛1₁⋯𝑡^𝑛𝑟_𝑟, (79) which is different from that in [6].

Remark 26. By Corollaries 8 and 9, the multivariable polynomials 𝐻_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟) satisfy 𝑖𝑥_𝑖𝜕𝜕𝑥_𝑖𝐻_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−𝑖+1,𝑛𝑖+1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=𝑛_𝑖−𝑖+1𝐻_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−𝑖+1,𝑛𝑖+1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟,_𝑟𝑖=1𝑖𝑥_𝑖𝜕𝜕𝑥_𝑖𝐻_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−𝑖+1,𝑛𝑖+1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^𝑟_𝑖=1𝑛_𝑖−𝑖+1×𝐻_𝑛^(𝛼)_{1,…,𝑛𝑖−1,𝑛𝑖−𝑖+1,𝑛𝑖+1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟. (80) for 𝑛_𝑖≥𝑖−1,    𝑖=1,2,…,𝑟; 𝑛_𝑗≥0, 𝑗≠𝑖.

Remark 27. From Theorem 11, the multivariable polynomials 𝐻_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟) have the following addition formula: 𝐻_𝑛^(𝛼+𝛽)_{1,…,𝑛𝑟}𝑥₁,…,𝑥_𝑟=^𝑛1_𝑘1=0⋯^𝑛𝑟_𝑘𝑟=0𝐻_𝑛^(𝛼)_{1−𝑘1,…,𝑛𝑟−𝑘𝑟}𝑥₁,…,𝑥_𝑟×𝐻_𝑘^(𝛽)_{1,…,𝑘𝑟}𝑥₁,…,𝑥_𝑟. (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_𝑛^(𝛼)_{1,…,𝑛𝑟}(𝑥₁,…,𝑥_𝑟).

Remark 28. If Ξ^𝐧,𝐩_{𝜆,𝜂,𝛼,𝛽,𝛾}𝑥₁,…,𝑥_𝑟;𝑢₁,…,𝑢_𝑟;𝐰∶=^{[𝑛1/𝑝1]}_𝑘1=0⋯^{[𝑛𝑟/𝑝𝑟]}_𝑘𝑟=0𝑎_{𝑘1,…,𝑘𝑟}𝐻_𝑛^(𝛼+𝛽)_{1−𝑝1𝑘1,…,𝑛𝑟−𝑝𝑟𝑘𝑟}𝑥₁,…,𝑥_𝑟×𝐻_𝜆^(𝛾)_{1+𝜂1𝑘1,…,𝜆𝑟+𝜂𝑟𝑘𝑟}𝑢₁,…,𝑢_𝑟𝑤^𝑘1₁⋯𝑤^𝑘𝑟_𝑟, (82) where 𝑎_{𝑘1,…,𝑘𝑟}≠0; 𝑝_𝑖∈ℕ; 𝑛_𝑖,𝜆_𝑖,𝜂_𝑖∈ℕ₀, 𝑖=1,2,…,𝑟, and 𝜆=(𝜆₁,…,𝜆_𝑟), 𝜂=(𝜂₁,…,𝜂_𝑟),𝐰=(𝑤₁,…,𝑤_𝑟), then we have ^𝑛1_𝑙1=0⋯^𝑛𝑟_𝑙𝑟=0^{[𝑙1/𝑝1]}_𝑘1=0⋯^{[𝑙𝑟/𝑝𝑟]}_𝑘𝑟=0𝑎_{𝑘1,…,𝑘𝑟}𝐻_𝑛^(𝛼)_{1−𝑙1,…,𝑛𝑟−𝑙𝑟}𝑥₁,…,𝑥_𝑟×𝐻_𝑙^(𝛽)_{1−𝑝1𝑘1,…,𝑙𝑟−𝑝𝑟𝑘𝑟}𝑥₁,…,𝑥_𝑟×𝐻_𝜆^(𝛾)_{1+𝜂1𝑘1,…,𝜆𝑟+𝜂𝑟𝑘𝑟}𝑢₁,…,𝑢_𝑟×𝑤^𝑘1₁⋯𝑤^𝑘𝑟_𝑟=Ξ^𝐧,𝐩_{𝜆,𝜂,𝛼,𝛽,𝛾}𝑥₁,…,𝑥_𝑟;𝑢₁,…,𝑢_𝑟;𝐰. (83)