# On a Family of Multivariate Modified Humbert Polynomials

Authors: Rabia Aktaş; Esra Erkuş-Duman
Journal: The Scientific World Journal
 (2013)

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## 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.


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## Body

## 1. Introduction

The generalized Humbert polynomials are generated by [1]
(1)(C-mxt+ytm)p=∑n=0∞‍Pn(m,x,y,p,C)tn,
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
(2)(1-∑i=1r‍(mixiti-yitimi))-α=∑n1,…,nr=0∞‍Θn1,…,nr(α)(x,y,m)t1n1⋯trnr,(|∑i=1r‍(mixiti-yitimi)|<1),
where x=(x1,…,xr),  y=(y1,…,yr), m=(m1,…,mr) and mi(i=1,2,…,r) is a positive integer. It follows from (2) that
(3)Θn1,…,nr(α)(x,y,m)=∑k1=0[n1/m1]‍⋯∑kr=0[nr/mr]‍(α)n1+⋯+nr-(m1-1)k1-⋯-(mr-1)krk1!⋯kr!(n1-m1k1)!⋯(nr-mrkr)!×(-1)k1+⋯+kr(m1x1)n1-m1k1⋯(mrxr)nr-mrkry1k1⋯yrkr,
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 functionΩμ(z) of s complex variables z1,…,zs(s∈ℕ) and of complex order μ, let
(4)Λμ,ν(z;w):=∑k=0∞‍akΩμ+νk(z)wk,
where (ak≠0,μ,ν∈ℂ); z=(z1,…,zs). Then, for p∈ℕ; x=(x1,…,xr); y=(y1,…,yr); m=(m1,…,mr), mi∈ℕ(i=1,2,…,r), one has
(5)∑n1,…,nr=0∞‍∑k=0[n1/p]‍akΘn1-pk,n2,…,nr(α)(x,y,m)×t2n2⋯trnrΩμ+νk(z)ηkt1n1-pk=(1-∑i=1r‍(mixiti-yitimi))-αΛμ,ν(z;η)
provided that each member of (5) exists.Proof.
For convenience, letS denote the first member of the assertion (5) of Theorem 1. Replacing n1 by n1+pk, we may write that
(6)S=∑n1,…,nr=0∞‍∑k=0∞‍akΘn1,n2,…,nr(α)(x,y,m)Ωμ+νk(z)ηkt1n1⋯trnr=∑n1,…,nr=0∞‍Θn1,n2,…,nr(α)(x,y,m)t1n1⋯trnr∑k=0∞‍akΩμ+νk(z)ηk=(1-∑i=1r‍(mixiti-yitimi))-αΛμ,ν(z;η),
which completes the proof.By using a similar idea, we also get the next result immediately.Theorem 2.
For a nonvanishing functionφλ1,…,λr(z) of s complex variables z1,…,zs  (s∈ℕ) and for pi∈ℕ, λi,ηi∈ℂ, z=(z1,…,zs),w=(w1,…,wr), λ:=(λ1,…,λr),  η:=(η1,…,ηr) let
(7)Ωλ,η,α,β,mn,p(x,y;z;w):=∑k1=0[n1/p1]‍⋯∑kr=0[nr/pr]‍ak1,…,krΘn1-p1k1,…,nr-prkr(α+β)(x,y,m)×φλ1+η1k1,…,λr+ηrkr(z)w1k1⋯wrkr,
where ak1,…,kr≠0; ni,(i=1,2,…,r)∈ℕ0; ℕ0:=ℕ∪{0}. Then, one has
(8)∑l1=0n1‍⋯∑lr=0nr‍∑k1=0[l1/p1]‍⋯∑kr=0[lr/pr]‍ak1,…,krΘn1-l1,…,nr-lr(α)(x,y,m)×Θl1-p1k1,…,lr-prkr(β)(x,y,m)×φλ1+η1k1,…,λr+ηrkr(z)×w1k1⋯wrkr=Ωλ,η,α,β,mn,p(x,y;z;w)
provided that each member of (8) exists.


## 3. Special Cases

As an application of the above theorems, when the multivariable functionΩμ+νk(z),  z=(z1,…,zs),  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
(9)s=r,Ωμ+νk(z)=gμ+νk(β1,…,βr)(z)
in Theorem 1, where the Chan-Chyan-Srivastava polynomials gμ+νk(β1,…,βr)(z) [10] are generated by
(10)∏i=1r‍(1-xit)-αi=∑n=0∞‍gn(α1,…,αr)(x1,…,xr)tn(αi∈ℂ(i=1,2,…,r);|t|<min{|x1|-1,…,|xr|-1}).
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.
IfΛμ,ν(z;w):=∑k=0∞akgμ+νk(β1,…,βr)(z)wk, ak≠0, μ,ν∈ℂ, z=(z1,…,zr) then, one has
(11)∑n1,…,nr=0∞‍∑k=0[n1/p]‍akΘn1-pk,n2,…,nr(α)(x,y,m)×t2n2⋯trnrgμ+νk(β1,…,βr)(z)ηkt1n1-pk=(1-∑i=1r‍(mixiti-yitimi))-αΛμ,ν(z;η)
provided that each member of (11) exists.Remark 4.
Using the generating relation (10) for the Chan-Chyan-Srivastava polynomials and getting ak=1,  μ=0, ν=1, we find that
(12)∑n1,…,nr=0∞‍∑k=0[n1/p]‍Θn1-pk,n2,…,nr(α)(x,y,m)t2n2⋯trnrgk(β1,…,βr)(z)×ηkt1n1-pk=(1-∑i=1r‍(mixiti-yitimi))-α∏i=1r‍(1-ziη)-βi,|η|<min{|z1|-1,…,|zr|-1},|∑i=1r‍(mixiti-yitimi)|<1.On the other hand, choosings=2rand
(13)φλ1+η1k1,…,λr+ηrkr(z)=Θλ1+η1k1,…,λr+ηrkr(γ)(u,v,m),λi,ηi(i=1,2,…,r)∈ℕ0,  u=(u1,…,ur), v=(v1,…,vr) 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(14)Λλ,η,α,β,γ,mn,p(x,y;u,v;w):=∑k1=0[n1/p1]‍⋯∑kr=0[nr/pr]‍ak1,…,krΘn1-p1k1,…,nr-prkr(α+β)(x,y,m)×Θλ1+η1k1,…,λr+ηrkr(γ)(u,v,m)w1k1⋯wrkr,
where ak1,…,kr≠0; pi∈ℕ; ni,λi,ηi∈ℕ0, i=1,2,…,r, then, we get
(15)∑l1=0n1‍⋯∑lr=0nr‍∑k1=0[l1/p1]‍⋯∑kr=0[lr/pr]‍ak1,…,krΘn1-l1,…,nr-lr(α)(x,y,m)×Θl1-p1k1,…,lr-prkr(β)(x,y,m)×Θλ1+η1k1,…,λr+ηrkr(γ)(u,v,m)×w1k1⋯wrkr=Λλ,η,α,β,γ,mn,p(x,y;u,v;w)
provided that each member of (15) exists.Furthermore, for every suitable choice of the coefficientsak(k∈ℕ0), if the multivariable functions Ωμ+νk(y) and φλ1+η1k1,…,λr+ηrkr(y),  y=(y1,…,ys),(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.
LetΨn1,…,nr(x,y,m) be a family of functions generated by
(16)G(m1x1t1-y1t1m1,…,mrxrtr-yrtrmr)=∑n1,…,nr=0∞‍Ψn1,…,nr(x,y,m)t1n1⋯trnr.
The following relations hold:
(17)niΨn1,…,nr(x,y,m)=xi∂∂xiΨn1,…,nr(x,y,m)-yi∂∂xiΨn1,…,ni-1,ni-mi+1,ni+1,…,nr(x,y,m)
for ni≥mi-1,  i=1,2,…,r; nj≥0, j≠i and
(18)niΨn1,…,nr(x,y,m)=xi∂∂xiΨn1,…,nr(x,y,m)
for ni=0,1,…,mi-2, i=1,2,…,r; nj≥0, j≠i. Also, one finds that
(19)mixi∂∂yiΨn1,…,nr(x,y,m)-miyi∂∂yi×Ψn1,…,ni-1,ni-mi+1,ni+1,…,nr(x,y,m)=-(ni-mi+1)Ψn1,…,ni-1,ni-mi+1,ni+1,…,nr(x,y,m)
for ni≥mi-1, i=1,2,…,r; nj≥0, j≠i and
(20)∂∂yiΨn1,…,nr(x,y,m)=0
for ni=0,1,…,mi-2,i=1,2,…,r; nj≥0, j≠i.Proof.
Fixi=1,2,…,r. Then, by differentiating (16) with respect to xi and ti, after making necessary calculations we obtain that
(21)∑n1,…,nr=0∞‍niΨn1,…,nr(x,y,m)t1n1⋯trnr=(xi-yitimi-1)∑n1,…,nr=0∞‍∂∂xiΨn1,…,nr(x,y,m)t1n1⋯trnr=xi∑n1,…,nr=0∞‍∂∂xiΨn1,…,nr(x,y,m)t1n1⋯trnr-yi∑n1,…,ni-1,ni+1,…,nr=0(ni≥mi-1)∞∂∂xiΨn1,…,ni-1,ni-mi+1,ni+1,…,nr(x,y,m)×t1n1⋯trnr.
Comparing the coefficients of t1n1⋯trnr, we obtain (17) and (18) for the fixed i. Similarly, if we differentiate (16) with respect to yi and ti, we can find the relation (19).Theorem 7.
IfΨn1,…,nr(x,y,m) is a family of functions generated by (16), then it satisfies the relations
(22)-mi∂∂yiΨn1,…,ni-1,ni-1,ni+1,…,nr(x,y,m)=∂∂xiΨn1,…,ni-1,ni-mi,ni+1,…,nr(x,y,m)
for ni≥mi, i=1,2,…,r; nj≥0, j≠i and
(23)∂∂yiΨn1,…,ni-1,ni-1,ni+1,…,nr(x,y,m)=0
for ni=1,2,…,mi-1, i=1,2,…,r; nj≥0, j≠i.Proof.
By comparing the derivatives of (16) with respect to xi and yi, we have
(24)-miti∑n1,…,nr=0∞‍∂∂yiΨn1,…,nr(x,y,m)t1n1⋯trnr=timi∑n1,…,nr=0∞‍∂∂xiΨn1,…,nr(x,y,m)t1n1⋯trnr,
which implies that
(25)-mi∂∂yiΨn1,…,ni-1,ni-1,ni+1,…,nr(x,y,m)=∂∂xiΨn1,…,ni-1,ni-mi,ni+1,…,nr(x,y,m)
for ni≥mi, i=1,2,…,r; nj≥0, j≠i and
(26)∂∂yiΨn1,…,ni-1,ni-1,ni+1,…,nr(x,y,m)=0
for ni=1,2,…,mi-1,i=1,2,…,r; nj≥0, j≠i. Thus, the proof is completed.As a result of these theorems, if we choose(27)G(m1x1t1-y1t1m1,…,mrxrtr-yrtrmr)=(1-∑i=1r‍(mixiti-yitimi))-α,
then we can give some recurrence relations for the family of multivariable polynomials given explicitly by (3). In the view of Theorem 6, we getCorollary 8.
For the family of multivariable polynomials generated by (2), the following relations hold
(28)niΘn1,…,nr(α)(x,y,m)=xi∂∂xiΘn1,…,nr(α)(x,y,m)-yi∂∂xiΘn1,…,ni-1,ni-mi+1,ni+1,…,nr(α)(x,y,m),(29)mixi∂∂yiΘn1,…,nr(α)(x,y,m)-miyi∂∂yi×Θn1,…,ni-1,ni-mi+1,ni+1,…,nr(α)(x,y,m)=-(ni-mi+1)Θn1,…,ni-1,ni-mi+1,ni+1,…,nr(α)(x,y,m)
for ni≥mi-1,i=1,2,…,r; nj≥0, j≠i. Also, for ni=0,1,…,mi-2, i=1,2,…,r; nj≥0, j≠i, we have
(30)niΘn1,…,nr(α)(x,y,m)=xi∂∂xiΘn1,…,nr(α)(x,y,m),(31)∂∂yiΘn1,…,nr(α)(x,y,m)=0.Corollary 9.
By summing the relations given by (28) and (29), respectively, for i=1,2,…,r, we get
(32)∑i=1r‍niΘn1,…,nr(α)(x,y,m)=∑i=1r‍xi∂∂xiΘn1,…,nr(α)(x,y,m)-∑i=1r‍yi∂∂xiΘn1,…,ni-1,ni-mi+1,ni+1,…,nr(α)(x,y,m),∑i=1r‍mixi∂∂yiΘn1,…,nr(α)(x,y,m)-∑i=1r‍miyi∂∂yiΘn1,…,ni-1,ni-mi+1,ni+1,…,nr(α)(x,y,m)=-∑i=1r‍(ni-mi+1)×Θn1,…,ni-1,ni-mi+1,ni+1,…,nr(α)(x,y,m)
for ni≥mi-1, i=1,2,…,r; nj≥0, j≠i.Similarly, as a consequence of Theorem7, we can give the next result at once.Corollary 10.
Other recurrence relations for the family of multivariable polynomialsΘn1,…,nr(α)(x,y,m) are
(33)-mi∂∂yiΘn1,…,ni-1,ni-1,ni+1,…,nr(α)(x,y,m)=∂∂xiΘn1,…,ni-1,ni-mi,ni+1,…,nr(α)(x,y,m)
for ni≥mi, i=1,2,…,r; nj≥0, j≠i and
(34)∂∂yiΘn1,…,ni-1,ni-1,ni+1,…,nr(α)(x,y,m)=0
for ni=1,2,…,mi-1, i=1,2,…,r; nj≥0, j≠i.Theorem 11.
The generating relation (2) yields the following addition formula for the family of multivariable polynomials given by (3)
(35)Θn1,…,nr(α+β)(x,y,m)=∑k1=0n1‍⋯∑kr=0nr‍Θn1-k1,…,nr-kr(α)(x,y,m)×Θk1,…,kr(β)(x,y,m).Proof.
From (2), we have
(36)∑n1,…,nr=0∞‍Θn1,…,nr(α+β)(x,y,m)t1n1⋯trnr=(1-∑i=1r‍(mixiti-yitimi))-(α+β)=(1-∑i=1r‍(mixiti-yitimi))-α×(1-∑i=1r‍(mixiti-yitimi))-β=∑n1,…,nr=0∞‍Θn1,…,nr(α)(x,y,m)t1n1⋯trnr×∑k1,…,kr=0∞‍Θk1,…,kr(β)(x,y,m)t1k1⋯trkr.
Replacing ni by ni-ki, i=1,2,…,r,   the right-hand side of the last equality is
(37)∑n1,…,nr=0∞‍∑k1=0n1‍⋯∑kr=0nr‍Θn1-k1,…,nr-kr(α)(x,y,m)×Θk1,…,kr(β)(x,y,m)t1n1⋯trnr.
Comparing the coefficients of t1n1⋯trnr completes the proof.We now give expansions of the family of multivariable polynomialsΘn1,…,nr(α)(x,y,m) given explicitly by (3) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials.Theorem 12.
Expansions ofΘn1,…,nr(α)(x,y,m) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are as follows
(38)Θn1,…,nr(α)(x,y,m)=∏i=1r‍{∑ki=0[ni/mi]‍∑si=0[(ni-miki)/2]‍(2ni-2miki-4si+1)ki!si!(3/2)ni-miki-si×(-1)kiyikiPni-miki-2si(mixi2)∑ki=0[ni/mi]‍}×(α)n1+⋯+nr-(m1-1)k1-⋯-(mr-1)kr,Θn1,…,nr(α)(x,y,m)=∏i=1r‍{∑ki=0[ni/mi]‍∑si=0[(ni-miki)/2]‍(νi+ni-miki-2si)ki!si!(νi)ni-miki-si+1×(-yi)kiCni-miki-2siνi(mixi2)∑ki=0[ni/mi]‍}×(α)n1+⋯+nr-(m1-1)k1-⋯-(mr-1)kr,Θn1,…,nr(α)(x,y,m)=∏i=1r‍{∑ki=0[ni/mi]‍∑si=0[(ni-miki)/2]‍(-yi)kiHni-miki-2si(mixi/2)ki!si!(ni-miki-2si)!}×(α)n1+⋯+nr-(m1-1)k1-⋯-(mr-1)kr,Θn1,…,nr(α)(x,y,m)=∏i=1r‍{∑ki=0[ni/mi]‍∑si=0ni-miki‍(βi+1)ni-miki2ni-miki(-1)siki!(ni-miki-si)!(βi+1)si×(-yi)kiLsi(βi)(mixi2)∑ki=0[ni/mi]‍}×(α)n1+⋯+nr-(m1-1)k1-⋯-(mr-1)kr.Proof.
By (2), we get
(39)∑n1,…,nr=0∞‍Θn1,…,nr(α)(x,y,m)t1n1⋯trnr=(1-∑i=1r‍(mixiti-yitimi))-α=∑n1,…,nr=0∞‍∑k1,…,kr=0∞‍(α)n1+⋯+nr+k1+⋯+krk1!⋯kr!n1!⋯nr!(-1)k1+⋯+kr×(m1x1)n1⋯(mrxr)nry1k1⋯yrkr×t1n1+m1k1⋯trnr+mrkr.
If we use the result in [11, page 181]
(40)(mx)nn!=∑s=0[n/2]‍(2n-4s+1)Pn-2s(mx/2)s!(3/2)n-s,
we can write that
(41)∑n1,…,nr=0∞‍Θn1,…,nr(α)(x,y,m)t1n1⋯trnr=∏i=1r‍{∑ni=0∞‍∑ki=0∞‍∑si=0[ni/2]‍(2ni-4si+1)si!ki!(3/2)ni-siPni-2si(mixi2)×(-yi)kitini+miki∑ni=0∞‍∑ki=0∞‍∑si=0[ni/2]‍(2ni-4si+1)si!ki!(3/2)ni-si}(α)n1+⋯+nr+k1+⋯+kr.
Getting ni-miki instead of ni in the last equality, we have
(42)∑n1,…,nr=0∞‍Θn1,…,nr(α)(x,y,m)t1n1⋯trnr=∑n1,…,nr=0∞‍∏i=1r‍{∑ki=0[ni/mi]‍∑si=0[(ni-miki)/2]‍(2ni-2miki-4si+1)si!ki!(3/2)ni-miki-si×Pni-miki-2si(mixi2)(-yi)kitini∑ki=0[ni/mi]‍}×(α)n1+⋯+nr-(m1-1)k1-⋯-(mr-1)kr=∑n1,…,nr=0∞‍{∏i=1r‍∑ki=0[ni/mi]‍∑si=0[(ni-miki)/2]‍(2ni-2miki-4si+1)si!ki!(3/2)ni-miki-si×Pni-miki-2si(mixi2)(-yi)ki∑ki=0[ni/mi]‍}×(α)n1+⋯+nr-(m1-1)k1-⋯-(mr-1)krt1n1⋯trnr.
Comparing the coefficients of t1n1⋯trnr 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)]
(43)(mx)nn!=∑k=0[n/2]‍(ν+n-2k)Cn-2kν(mx/2)k!(ν)n+1-k,(mx)nn!=∑k=0[n/2]‍Hn-2k(mx/2)k!(n-2k)!,(mx)nn!=2n∑k=0n‍(-1)k(α+1)nLk(α)(mx/2)(n-k)!(α+1)k,
one can easily obtain the other expansions of Θn1,…,nr(α)(x,y,m) in series of Gegenbauer, Hermite, and Laguerre polynomials.


## 5. The Special Cases ofΘn(α) (x, y, m) and Some Properties

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


### 5.1. The Case ofmi=2, yi=1, i=1,2,…,r in (2)

This case gives a multivariable analogue of Gegenbauer polynomials(44)(1-2x1t1+t12-⋯-2xrtr+tr2)-α=∑n1,…,nr=0∞‍Cn1,…,nr(α)(x1,…,xr)t1n1⋯trnr,
which reduces to two variables Gegenbauer polynomials given by [12] for r=2. Equation (44) yields the following formula:
(45)Cn1,…,nr(α)(x1,…,xr)=∑k1=0[n1/2]‍⋯∑kr=0[nr/2]‍(⋯(nr-2kr)!∫)-1(∫(α)n1+⋯+nr-k1-⋯-kr)×(k1!⋯kr!(n1-2k1)!⋯(nr-2kr)!∫)-1)×(-1)k1+⋯+kr(2x1)n1-2k1⋯(2xr)nr-2kr.It follows thatCn1,…,nr(α)(x1,…,xr) is a polynomial of degree ni with respect to the fixed variable xi(i=1,2,…,r).   Thus, Cn1,…,nr(α)(x1,…,xr) is a polynomial of total degree (n1+⋯+nr) with respect to the variables x1,…,xr. Equation (45) also yields
(46)Cn1,…,nr(α)(x1,…,xr)=2n1+⋯+nr(α)n1+⋯+nrn1!⋯nr!x1n1⋯xrnr+Π(x1,…,xr),
where Π(x1,…,xr) is a polynomial of degree (n1+⋯+nr-2) with respect to the variables x1,…,xr. In (44), by getting x1→-x1 and t1→-t1, we have
(47)Cn1,…,nr(α)(-x1,x2,…,xr)=(-1)n1Cn1,…,nr(α)(x1,…,xr).Similarly, fori=1,2,…,r,   we get
(48)Cn1,…,nr(α)(x1,…,xi-1,-xi,xi+1,…,xr)=(-1)niCn1,…,nr(α)(x1,…,xr).Takingxi→-xi andti→-ti, i=1,2,…,r, in (44), we obtain
(49)Cn1,…,nr(α)(-x1,…,-xr)=(-1)n1+⋯+nrCn1,…,nr(α)(x1,…,xr).Theorem 13.
For the polynomialsCn1,…,nr(α)(x1,…,xr), one has
(50)C2n1,…,2nr(α)(0,0,…,0)=(-1)n1+⋯+nr(α)n1+⋯+nrn1!⋯nr!
and if at least one of ni, i=1,2,…,r, is odd; then
(51)Cn1,….,nr(α)(0,0,…,0)=0.Proof.
If we set allxi=0, i=1,2,…,r in (44), we have
(52)(1+t12+⋯+tr2)-α=∑n1,…,nr=0∞‍Cn1,…,nr(α)(0,0,…,0)t1n1⋯trnr.
On the other hand, we get
(53)(1+t12+⋯+tr2)-α=∑n1,…,nr=0∞‍(-1)n1+⋯+nrn1!⋯nr!(α)n1+⋯+nrt12n1⋯tr2nr.
By comparing the coefficients of t1n1⋯trnr, we obtain the desired.From the theorems and corollaries given in Section4, we can give some other properties of Cn1,…,nr(α)(x1,…,xr).Remark 14.
By Corollaries8 and 9, for the family of multivariable polynomials generated by (44), the following relations:
(54)niCn1,…,nr(α)(x1,…,xr)=xi∂∂xiCn1,…,nr(α)(x1,…,xr)-∂∂xiCn1,…,ni-1,ni-1,ni+1,…,nr(α)(x1,…,xr),∑i=1r‍niCn1,…,nr(α)(x1,…,xr)=∑i=1r‍xi∂∂xiCn1,…,nr(α)(x1,…,xr)-∑i=1r‍∂∂xiCn1,…,ni-1,ni-1,ni+1,…,nr(α)(x1,…,xr)
hold for ni≥1,i=1,2,…,r.Remark 15.
From Theorem11, the multivariable polynomials Cn1,…,nr(α)(x1,…,xr) satisfy the following addition formula:
(55)Cn1,…,nr(α+β)(x1,…,xr)=∑k1=0n1‍⋯∑kr=0nr‍Cn1-k1,…,nr-kr(α)(x1,…,xr)×Ck1,…,kr(β)(x1,…,xr).Remark 16.
As a result of Theorem12, expansions of Cn1,…,nr(α)(x1,…,xr) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are as follows:
(56)Cn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑ki=0[ni/2]‍∑si=0[(ni-2ki)/2]‍(2ni-4ki-4si+1)ki!si!(3/2)ni-2ki-si×(-1)kiPni-2ki-2si(xi)∑ki=0[ni/2]‍}×(α)n1+⋯+nr-k1-⋯-kr,Cn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑ki=0[ni/2]‍∑si=0[(ni-2ki)/2]‍(νi+ni-2ki-2si)ki!si!(νi)ni-2ki-si+1×(-1)kiCni-2ki-2siνi(xi)∑ki=0[ni/2]‍}×(α)n1+⋯+nr-k1-⋯-kr,Cn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑ki=0[ni/2]‍∑si=0[(ni-2ki)/2]‍(-1)kiHni-2ki-2si(xi)ki!si!(ni-2ki-2si)!}×(α)n1+⋯+nr-k1-⋯-kr,Cn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑ki=0[ni/2]‍∑si=0ni-2ki‍(βi+1)ni-2ki2ni-2ki(-1)siki!(ni-2ki-si)!(βi+1)si×(-1)kiLsi(βi)(xi)∑ki=0[ni/2]‍}×(α)n1+⋯+nr-k1-⋯-kr.We now give a hypergeometric representation for the multivariable polynomialsCn1,…,nr(α)(x1,…,xr) given by (44).Theorem 17.
The multivariable polynomialsCn1,…,nr(α)(x1,…,xr) have the following hypergeometric representation:
(57)Cn1,…,nr(α)(x1,…,xr)=(α)n1+⋯+nrn1!⋯nr!(2x1)n1⋯(2xr)nr×FB(r)[1xr2-n12,…,-nr2,-n1+12,…,-nr+12;1-α-n1-⋯-nr;1x12,…,1xr2],(max{1x12,…,1xr2}<1),
where FB(r) is a Lauricella function of r-variable defined by [13] (see also [2])
(58)FB(r)[a1,…,ar,b1,…,br;c;x1,…,xr]=∑n1,…,nr=0∞‍(a1)n1⋯(ar)nr(b1)n1⋯(br)nr(c)n1+⋯+nrx1n1n1!⋯xrnrnr!,(max{|x1|,…,|xr|}<1).Proof.
We have the following results from [11]:
(59)(α)n-k=(-1)k(α)n(1-α-n)k,(-n)2k=22k(-n2)k(-n+12)k,(n-2k)!=n!(-n)2k.
In view of these relations, the equality (45) can be written as follows:
(60)Cn1,…,nr(α)(x1,…,xr)=∑k1=0[n1/2]‍⋯∑kr=0[nr/2](×(1-α-n1-⋯-nr)k1+⋯+kr((α)n1+⋯+nr22(k1+⋯+kr)(-n12)k1)-1((α)n1+⋯+nr22(k1+⋯+kr)(-n12)k1‍×(-n1+12)k1⋯(-nr2)kr×(-nr+12)kr)×(((α)n1+⋯+nr22(k1+⋯+kr)(-n12)k1k1!⋯kr!n1!⋯nr!×(1-α-n1-⋯-nr)k1+⋯+kr((α)n1+⋯+nr22(k1+⋯+kr)(-n12)k1)-1)×(2x1)n1-2k1⋯(2xr)nr-2kr=(α)n1+⋯+nrn1!⋯nr!(2x1)n1⋯(2xr)nr×∑k1=0[n1/2]‍⋯∑kr=0[nr/2]‍(∑k1=0[n12]‍((-n12)k1(-n1+12)k1⋯(-nr2)kr(-nr+12)kr)×(((-n12)k1(-n1+12)k1)(1-α-n1-⋯-nr)k1+⋯+kr×k1!⋯kr!((-n12)k1(-n1+12)k1))-1∑k1=0[n12]‍)×(1x12)k1⋯(1xr2)kr=FB(r)[1xr2-n12,…,-nr2,-n1+12,…,-nr+12;1-α-n1-⋯-nr;1x12,…,1xr2]×(α)n1+⋯+nrn1!⋯nr!(2x1)n1⋯(2xr)nr,
where max{1/x12,…,1/xr2}<1. The proof is completed.Remark 18.
Forr=2,Theorem 17 reduces to the known result for two variable Gegenbauer polynomials given by [12].


### 5.2. The Case ofxi=0, yi=-xi, i=1,2,…,r in (2)

This case yields(61)(1-x1t1m1-x2t2m2-⋯-xrtrmr)-α=∑n1,…,nr=0∞‍Un1,…,nr(α)(x1,…,xr)t1n1⋯trnr,
which is a different unification from that in [8].Remark 19.
From Corollaries8 and 9, the multivariable polynomials Un1,…,nr(α)(x1,…,xr) satisfy
(62)mixi∂∂xiUn1,…,ni-1,ni-mi+1,ni+1,…,nr(α)(x1,…,xr)=(ni-mi+1)Un1,…,ni-1,ni-mi+1,ni+1,…,nr(α)(x1,…,xr),∑i=1r‍mixi∂∂xi×Un1,…,ni-1,ni-mi+1,ni+1,…,nr(α)(x1,…,xr)=∑i=1r‍(ni-mi+1)×Un1,…,ni-1,ni-mi+1,ni+1,…,nr(α)(x1,…,xr).
for ni≥mi-1, i=1,2,…,r; nj≥0, j≠i.Remark 20.
As result of Theorem11, the multivariable polynomials Un1,…,nr(α)(x1,…,xr) have the following addition formula:
(63)Un1,…,nr(α+β)(x1,…,xr)=∑k1=0n1‍⋯∑kr=0nr‍Un1-k1,…,nr-kr(α)(x1,…,xr)×Uk1,…,kr(β)(x1,…,xr).


#### 5.2.1. The Case ofmi=1, i=1,2,…,r in Section 5.2

In this case, we get ar-variable analogue of Lagrange polynomials (see [14]) which is different from that in [10]
(64)(1-x1t1-⋯-xrtr)-α=∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)t1n1⋯trnr.They are given explicitly by(65)Gn1,…,nr(α)(x1,…,xr)=(α)n1+⋯+nrn1!⋯nr!x1n1⋯xrnr.Remark 21.
From Corollaries8 and 9, the multivariable polynomials Gn1,…,nr(α)(x1,…,xr) satisfy
(66)niGn1,…,nr(α)(x1,…,xr)=xi∂∂xiGn1,…,nr(α)(x1,…,xr)
for i=1,2,…,r and
(67)∑i=1r‍niGn1,…,nr(α)(x1,…,xr)=∑i=1r‍xi∂∂xiGn1,…,nr(α)(x1,…,xr).Remark 22.
By Theorem11, the multivariable polynomials Gn1,…,nr(α)(x1,…,xr) have the following addition formula:
(68)Gn1,…,nr(α+β)(x1,…,xr)=∑k1=0n1‍⋯∑kr=0nr‍Gn1-k1,…,nr-kr(α)(x1,…,xr)×Gk1,…,kr(β)(x1,…,xr).Remark 23.
From Theorem12, expansions of Gn1,…,nr(α)(x1,…,xr) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are given by
(69)Gn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑si=0[ni/2]‍(2ni-4si+1)si!(3/2)ni-siPni-2si(xi2)}×(α)n1+⋯+nr,Gn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑si=0[ni/2]‍(νi+ni-2si)si!(νi)ni-si+1Cni-2siνi(xi2)}×(α)n1+⋯+nr,Gn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑si=0[ni/2]‍Hni-2si(xi/2)si!(ni-2si)!}(α)n1+⋯+nr,Gn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑si=0ni‍(βi+1)ni2ni(-1)si(ni-si)!(βi+1)siLsi(βi)(xi2)}×(α)n1+⋯+nr.We can discuss some generating functions for the multivariable polynomialsGn1,…,nr(α)(x1,…,xr).Theorem 24.
For the polynomialsGn1,…,nr(α)(x1,…,xr),   the following generating function holds true for k∈ℕ0:
(70)∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)[k+ni]mz1n1⋯zrnr=m!(1-x1z1-⋯-xrzr)-α×gm(α,-k)(xizi1-x1z1-⋯-xrzr,-1),
for each i=1,2,…,r, where
(71)[k]m:=k(k-1)⋯(k-m+1),m∈ℕ,[k]0=1
and gn(α,β)(x,y) denotes the classical Lagrange polynomials defined by [14]
(72)(1-xt)-α(1-yt)-β=∑n=0∞‍gn(α,β)(x,y)tn(α,β∈ℂ;|t|<min{|x|-1,|y|-1}).Proof.
Fixi=1,2,…,r. Let 𝒞ξiand 𝒞ξi* denote the circles of radius ɛi, centered ξi=zi and ξi=0, respectively, where
(73)0<ɛi<|xi|-1|1-∑k=1r‍xkzk|.
Here these circles are described in the positive direction. By Cauchy's Integral Formula, we have
(74)Dzim{(1-x1z1-⋯-xrzr)-αzik}=m!2πi∮𝒞ξiξik(1-x1z1-⋯-xiξi-⋯-xrzr)-α(ξi-zi)m+1dξi=m!2πi×∮𝒞ξi*(ξi+zi)k(1-x1z1-⋯-xi(ξi+zi)-⋯-xrzr)-αξim+1dξi=m!2πizik(1-x1z1-⋯-xrzr)-α×∮𝒞ξi*(1+ξi/zi)kξim+1(1-xiξi1-x1z1-⋯-xrzr)-αdξi=m!zik(1-x1z1-⋯-xrzr)-α×gm(α,-k)(xi1-x1z1-⋯-xrzr,-1zi)=m!zik-m(1-x1z1-⋯-xrzr)-α×gm(α,-k)(xizi1-x1z1-⋯-xrzr,-1).
On the other hand, we can write that
(75)Dzim{(1-x1z1-⋯-xrzr)-αzik}=Dzim{zik∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)z1n1⋯zrnr}=∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)dmdzimz1n1⋯zini+k⋯zrnr=zik-m∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)[k+ni]mz1n1⋯zrnr.
From (74) and (75), we see that
(76)∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)[k+ni]mz1n1⋯zrnr=m!(1-x1z1-⋯-xrzr)-α×gm(α,-k)(xizi1-x1z1-⋯-xrzr,-1).Corollary 25.
From (71), one observes that
(77)[ni+m]m=(ni+m)(ni+m-1)⋯(ni+1)=(ni+1)m
for each i=1,2,…,r. By setting k=min (70) and then using (77), one has
(78)∑n1,…,nr=0∞‍(ni+1)mGn1,…,nr(α)(x1,…,xr)z1n1⋯zrnr=m!(1-x1z1-⋯-xrzr)-α×gm(α,-m)(xizi1-x1z1-⋯-xrzr,-1).


#### 5.2.2. The Case ofmi=i,i=1,2,…,r in Section 5.2

This case reduces to a multivariable analogue of Lagrange-Hermite polynomials(79)(1-x1t1-x2t22-⋯-xrtrr)-α=∑n1,…,nr=0∞‍Hn1,…,nr(α)(x1,…,xr)t1n1⋯trnr,
which is different from that in [6].Remark 26.
By Corollaries8 and 9, the multivariable polynomials Hn1,…,nr(α)(x1,…,xr) satisfy
(80)ixi∂∂xiHn1,…,ni-1,ni-i+1,ni+1,…,nr(α)(x1,…,xr)=(ni-i+1)Hn1,…,ni-1,ni-i+1,ni+1,…,nr(α)(x1,…,xr),∑i=1r‍ixi∂∂xiHn1,…,ni-1,ni-i+1,ni+1,…,nr(α)(x1,…,xr)=∑i=1r‍(ni-i+1)×Hn1,…,ni-1,ni-i+1,ni+1,…,nr(α)(x1,…,xr).
for ni≥i-1,    i=1,2,…,r; nj≥0, j≠i.Remark 27.
From Theorem11, the multivariable polynomials Hn1,…,nr(α)(x1,…,xr) have the following addition formula:
(81)Hn1,…,nr(α+β)(x1,…,xr)=∑k1=0n1‍⋯∑kr=0nr‍Hn1-k1,…,nr-kr(α)(x1,…,xr)×Hk1,…,kr(β)(x1,…,xr).Furthermore, settingmi=i, xi=0, yi=-xi, i=1,2,…,r in Corollary 5, we obtain the following class of bilinear generating functions for the polynomials Hn1,…,nr(α)(x1,…,xr).Remark 28.
If(82)Ξλ,η,α,β,γn,p(x1,…,xr;u1,…,ur;w):=∑k1=0[n1/p1]‍⋯∑kr=0[nr/pr]‍ak1,…,krHn1-p1k1,…,nr-prkr(α+β)(x1,…,xr)×Hλ1+η1k1,…,λr+ηrkr(γ)(u1,…,ur)w1k1⋯wrkr,
where ak1,…,kr≠0; pi∈ℕ; ni,λi,ηi∈ℕ0, i=1,2,…,r, and λ=(λ1,…,λr), η=(η1,…,ηr),w=(w1,…,wr), then we have
(83)∑l1=0n1‍⋯∑lr=0nr‍∑k1=0[l1/p1]‍⋯∑kr=0[lr/pr]‍ak1,…,krHn1-l1,…,nr-lr(α)(x1,…,xr)×Hl1-p1k1,…,lr-prkr(β)(x1,…,xr)×Hλ1+η1k1,…,λr+ηrkr(γ)(u1,…,ur)×w1k1⋯wrkr=Ξλ,η,α,β,γn,p(x1,…,xr;u1,…,ur;w).


## 5.1. The Case ofmi=2, yi=1, i=1,2,…,r in (2)

This case gives a multivariable analogue of Gegenbauer polynomials(44)(1-2x1t1+t12-⋯-2xrtr+tr2)-α=∑n1,…,nr=0∞‍Cn1,…,nr(α)(x1,…,xr)t1n1⋯trnr,
which reduces to two variables Gegenbauer polynomials given by [12] for r=2. Equation (44) yields the following formula:
(45)Cn1,…,nr(α)(x1,…,xr)=∑k1=0[n1/2]‍⋯∑kr=0[nr/2]‍(⋯(nr-2kr)!∫)-1(∫(α)n1+⋯+nr-k1-⋯-kr)×(k1!⋯kr!(n1-2k1)!⋯(nr-2kr)!∫)-1)×(-1)k1+⋯+kr(2x1)n1-2k1⋯(2xr)nr-2kr.It follows thatCn1,…,nr(α)(x1,…,xr) is a polynomial of degree ni with respect to the fixed variable xi(i=1,2,…,r).   Thus, Cn1,…,nr(α)(x1,…,xr) is a polynomial of total degree (n1+⋯+nr) with respect to the variables x1,…,xr. Equation (45) also yields
(46)Cn1,…,nr(α)(x1,…,xr)=2n1+⋯+nr(α)n1+⋯+nrn1!⋯nr!x1n1⋯xrnr+Π(x1,…,xr),
where Π(x1,…,xr) is a polynomial of degree (n1+⋯+nr-2) with respect to the variables x1,…,xr. In (44), by getting x1→-x1 and t1→-t1, we have
(47)Cn1,…,nr(α)(-x1,x2,…,xr)=(-1)n1Cn1,…,nr(α)(x1,…,xr).Similarly, fori=1,2,…,r,   we get
(48)Cn1,…,nr(α)(x1,…,xi-1,-xi,xi+1,…,xr)=(-1)niCn1,…,nr(α)(x1,…,xr).Takingxi→-xi andti→-ti, i=1,2,…,r, in (44), we obtain
(49)Cn1,…,nr(α)(-x1,…,-xr)=(-1)n1+⋯+nrCn1,…,nr(α)(x1,…,xr).Theorem 13.
For the polynomialsCn1,…,nr(α)(x1,…,xr), one has
(50)C2n1,…,2nr(α)(0,0,…,0)=(-1)n1+⋯+nr(α)n1+⋯+nrn1!⋯nr!
and if at least one of ni, i=1,2,…,r, is odd; then
(51)Cn1,….,nr(α)(0,0,…,0)=0.Proof.
If we set allxi=0, i=1,2,…,r in (44), we have
(52)(1+t12+⋯+tr2)-α=∑n1,…,nr=0∞‍Cn1,…,nr(α)(0,0,…,0)t1n1⋯trnr.
On the other hand, we get
(53)(1+t12+⋯+tr2)-α=∑n1,…,nr=0∞‍(-1)n1+⋯+nrn1!⋯nr!(α)n1+⋯+nrt12n1⋯tr2nr.
By comparing the coefficients of t1n1⋯trnr, we obtain the desired.From the theorems and corollaries given in Section4, we can give some other properties of Cn1,…,nr(α)(x1,…,xr).Remark 14.
By Corollaries8 and 9, for the family of multivariable polynomials generated by (44), the following relations:
(54)niCn1,…,nr(α)(x1,…,xr)=xi∂∂xiCn1,…,nr(α)(x1,…,xr)-∂∂xiCn1,…,ni-1,ni-1,ni+1,…,nr(α)(x1,…,xr),∑i=1r‍niCn1,…,nr(α)(x1,…,xr)=∑i=1r‍xi∂∂xiCn1,…,nr(α)(x1,…,xr)-∑i=1r‍∂∂xiCn1,…,ni-1,ni-1,ni+1,…,nr(α)(x1,…,xr)
hold for ni≥1,i=1,2,…,r.Remark 15.
From Theorem11, the multivariable polynomials Cn1,…,nr(α)(x1,…,xr) satisfy the following addition formula:
(55)Cn1,…,nr(α+β)(x1,…,xr)=∑k1=0n1‍⋯∑kr=0nr‍Cn1-k1,…,nr-kr(α)(x1,…,xr)×Ck1,…,kr(β)(x1,…,xr).Remark 16.
As a result of Theorem12, expansions of Cn1,…,nr(α)(x1,…,xr) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are as follows:
(56)Cn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑ki=0[ni/2]‍∑si=0[(ni-2ki)/2]‍(2ni-4ki-4si+1)ki!si!(3/2)ni-2ki-si×(-1)kiPni-2ki-2si(xi)∑ki=0[ni/2]‍}×(α)n1+⋯+nr-k1-⋯-kr,Cn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑ki=0[ni/2]‍∑si=0[(ni-2ki)/2]‍(νi+ni-2ki-2si)ki!si!(νi)ni-2ki-si+1×(-1)kiCni-2ki-2siνi(xi)∑ki=0[ni/2]‍}×(α)n1+⋯+nr-k1-⋯-kr,Cn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑ki=0[ni/2]‍∑si=0[(ni-2ki)/2]‍(-1)kiHni-2ki-2si(xi)ki!si!(ni-2ki-2si)!}×(α)n1+⋯+nr-k1-⋯-kr,Cn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑ki=0[ni/2]‍∑si=0ni-2ki‍(βi+1)ni-2ki2ni-2ki(-1)siki!(ni-2ki-si)!(βi+1)si×(-1)kiLsi(βi)(xi)∑ki=0[ni/2]‍}×(α)n1+⋯+nr-k1-⋯-kr.We now give a hypergeometric representation for the multivariable polynomialsCn1,…,nr(α)(x1,…,xr) given by (44).Theorem 17.
The multivariable polynomialsCn1,…,nr(α)(x1,…,xr) have the following hypergeometric representation:
(57)Cn1,…,nr(α)(x1,…,xr)=(α)n1+⋯+nrn1!⋯nr!(2x1)n1⋯(2xr)nr×FB(r)[1xr2-n12,…,-nr2,-n1+12,…,-nr+12;1-α-n1-⋯-nr;1x12,…,1xr2],(max{1x12,…,1xr2}<1),
where FB(r) is a Lauricella function of r-variable defined by [13] (see also [2])
(58)FB(r)[a1,…,ar,b1,…,br;c;x1,…,xr]=∑n1,…,nr=0∞‍(a1)n1⋯(ar)nr(b1)n1⋯(br)nr(c)n1+⋯+nrx1n1n1!⋯xrnrnr!,(max{|x1|,…,|xr|}<1).Proof.
We have the following results from [11]:
(59)(α)n-k=(-1)k(α)n(1-α-n)k,(-n)2k=22k(-n2)k(-n+12)k,(n-2k)!=n!(-n)2k.
In view of these relations, the equality (45) can be written as follows:
(60)Cn1,…,nr(α)(x1,…,xr)=∑k1=0[n1/2]‍⋯∑kr=0[nr/2](×(1-α-n1-⋯-nr)k1+⋯+kr((α)n1+⋯+nr22(k1+⋯+kr)(-n12)k1)-1((α)n1+⋯+nr22(k1+⋯+kr)(-n12)k1‍×(-n1+12)k1⋯(-nr2)kr×(-nr+12)kr)×(((α)n1+⋯+nr22(k1+⋯+kr)(-n12)k1k1!⋯kr!n1!⋯nr!×(1-α-n1-⋯-nr)k1+⋯+kr((α)n1+⋯+nr22(k1+⋯+kr)(-n12)k1)-1)×(2x1)n1-2k1⋯(2xr)nr-2kr=(α)n1+⋯+nrn1!⋯nr!(2x1)n1⋯(2xr)nr×∑k1=0[n1/2]‍⋯∑kr=0[nr/2]‍(∑k1=0[n12]‍((-n12)k1(-n1+12)k1⋯(-nr2)kr(-nr+12)kr)×(((-n12)k1(-n1+12)k1)(1-α-n1-⋯-nr)k1+⋯+kr×k1!⋯kr!((-n12)k1(-n1+12)k1))-1∑k1=0[n12]‍)×(1x12)k1⋯(1xr2)kr=FB(r)[1xr2-n12,…,-nr2,-n1+12,…,-nr+12;1-α-n1-⋯-nr;1x12,…,1xr2]×(α)n1+⋯+nrn1!⋯nr!(2x1)n1⋯(2xr)nr,
where max{1/x12,…,1/xr2}<1. The proof is completed.Remark 18.
Forr=2,Theorem 17 reduces to the known result for two variable Gegenbauer polynomials given by [12].


## 5.2. The Case ofxi=0, yi=-xi, i=1,2,…,r in (2)

This case yields(61)(1-x1t1m1-x2t2m2-⋯-xrtrmr)-α=∑n1,…,nr=0∞‍Un1,…,nr(α)(x1,…,xr)t1n1⋯trnr,
which is a different unification from that in [8].Remark 19.
From Corollaries8 and 9, the multivariable polynomials Un1,…,nr(α)(x1,…,xr) satisfy
(62)mixi∂∂xiUn1,…,ni-1,ni-mi+1,ni+1,…,nr(α)(x1,…,xr)=(ni-mi+1)Un1,…,ni-1,ni-mi+1,ni+1,…,nr(α)(x1,…,xr),∑i=1r‍mixi∂∂xi×Un1,…,ni-1,ni-mi+1,ni+1,…,nr(α)(x1,…,xr)=∑i=1r‍(ni-mi+1)×Un1,…,ni-1,ni-mi+1,ni+1,…,nr(α)(x1,…,xr).
for ni≥mi-1, i=1,2,…,r; nj≥0, j≠i.Remark 20.
As result of Theorem11, the multivariable polynomials Un1,…,nr(α)(x1,…,xr) have the following addition formula:
(63)Un1,…,nr(α+β)(x1,…,xr)=∑k1=0n1‍⋯∑kr=0nr‍Un1-k1,…,nr-kr(α)(x1,…,xr)×Uk1,…,kr(β)(x1,…,xr).


### 5.2.1. The Case ofmi=1, i=1,2,…,r in Section 5.2

In this case, we get ar-variable analogue of Lagrange polynomials (see [14]) which is different from that in [10]
(64)(1-x1t1-⋯-xrtr)-α=∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)t1n1⋯trnr.They are given explicitly by(65)Gn1,…,nr(α)(x1,…,xr)=(α)n1+⋯+nrn1!⋯nr!x1n1⋯xrnr.Remark 21.
From Corollaries8 and 9, the multivariable polynomials Gn1,…,nr(α)(x1,…,xr) satisfy
(66)niGn1,…,nr(α)(x1,…,xr)=xi∂∂xiGn1,…,nr(α)(x1,…,xr)
for i=1,2,…,r and
(67)∑i=1r‍niGn1,…,nr(α)(x1,…,xr)=∑i=1r‍xi∂∂xiGn1,…,nr(α)(x1,…,xr).Remark 22.
By Theorem11, the multivariable polynomials Gn1,…,nr(α)(x1,…,xr) have the following addition formula:
(68)Gn1,…,nr(α+β)(x1,…,xr)=∑k1=0n1‍⋯∑kr=0nr‍Gn1-k1,…,nr-kr(α)(x1,…,xr)×Gk1,…,kr(β)(x1,…,xr).Remark 23.
From Theorem12, expansions of Gn1,…,nr(α)(x1,…,xr) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are given by
(69)Gn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑si=0[ni/2]‍(2ni-4si+1)si!(3/2)ni-siPni-2si(xi2)}×(α)n1+⋯+nr,Gn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑si=0[ni/2]‍(νi+ni-2si)si!(νi)ni-si+1Cni-2siνi(xi2)}×(α)n1+⋯+nr,Gn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑si=0[ni/2]‍Hni-2si(xi/2)si!(ni-2si)!}(α)n1+⋯+nr,Gn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑si=0ni‍(βi+1)ni2ni(-1)si(ni-si)!(βi+1)siLsi(βi)(xi2)}×(α)n1+⋯+nr.We can discuss some generating functions for the multivariable polynomialsGn1,…,nr(α)(x1,…,xr).Theorem 24.
For the polynomialsGn1,…,nr(α)(x1,…,xr),   the following generating function holds true for k∈ℕ0:
(70)∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)[k+ni]mz1n1⋯zrnr=m!(1-x1z1-⋯-xrzr)-α×gm(α,-k)(xizi1-x1z1-⋯-xrzr,-1),
for each i=1,2,…,r, where
(71)[k]m:=k(k-1)⋯(k-m+1),m∈ℕ,[k]0=1
and gn(α,β)(x,y) denotes the classical Lagrange polynomials defined by [14]
(72)(1-xt)-α(1-yt)-β=∑n=0∞‍gn(α,β)(x,y)tn(α,β∈ℂ;|t|<min{|x|-1,|y|-1}).Proof.
Fixi=1,2,…,r. Let 𝒞ξiand 𝒞ξi* denote the circles of radius ɛi, centered ξi=zi and ξi=0, respectively, where
(73)0<ɛi<|xi|-1|1-∑k=1r‍xkzk|.
Here these circles are described in the positive direction. By Cauchy's Integral Formula, we have
(74)Dzim{(1-x1z1-⋯-xrzr)-αzik}=m!2πi∮𝒞ξiξik(1-x1z1-⋯-xiξi-⋯-xrzr)-α(ξi-zi)m+1dξi=m!2πi×∮𝒞ξi*(ξi+zi)k(1-x1z1-⋯-xi(ξi+zi)-⋯-xrzr)-αξim+1dξi=m!2πizik(1-x1z1-⋯-xrzr)-α×∮𝒞ξi*(1+ξi/zi)kξim+1(1-xiξi1-x1z1-⋯-xrzr)-αdξi=m!zik(1-x1z1-⋯-xrzr)-α×gm(α,-k)(xi1-x1z1-⋯-xrzr,-1zi)=m!zik-m(1-x1z1-⋯-xrzr)-α×gm(α,-k)(xizi1-x1z1-⋯-xrzr,-1).
On the other hand, we can write that
(75)Dzim{(1-x1z1-⋯-xrzr)-αzik}=Dzim{zik∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)z1n1⋯zrnr}=∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)dmdzimz1n1⋯zini+k⋯zrnr=zik-m∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)[k+ni]mz1n1⋯zrnr.
From (74) and (75), we see that
(76)∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)[k+ni]mz1n1⋯zrnr=m!(1-x1z1-⋯-xrzr)-α×gm(α,-k)(xizi1-x1z1-⋯-xrzr,-1).Corollary 25.
From (71), one observes that
(77)[ni+m]m=(ni+m)(ni+m-1)⋯(ni+1)=(ni+1)m
for each i=1,2,…,r. By setting k=min (70) and then using (77), one has
(78)∑n1,…,nr=0∞‍(ni+1)mGn1,…,nr(α)(x1,…,xr)z1n1⋯zrnr=m!(1-x1z1-⋯-xrzr)-α×gm(α,-m)(xizi1-x1z1-⋯-xrzr,-1).


### 5.2.2. The Case ofmi=i,i=1,2,…,r in Section 5.2

This case reduces to a multivariable analogue of Lagrange-Hermite polynomials(79)(1-x1t1-x2t22-⋯-xrtrr)-α=∑n1,…,nr=0∞‍Hn1,…,nr(α)(x1,…,xr)t1n1⋯trnr,
which is different from that in [6].Remark 26.
By Corollaries8 and 9, the multivariable polynomials Hn1,…,nr(α)(x1,…,xr) satisfy
(80)ixi∂∂xiHn1,…,ni-1,ni-i+1,ni+1,…,nr(α)(x1,…,xr)=(ni-i+1)Hn1,…,ni-1,ni-i+1,ni+1,…,nr(α)(x1,…,xr),∑i=1r‍ixi∂∂xiHn1,…,ni-1,ni-i+1,ni+1,…,nr(α)(x1,…,xr)=∑i=1r‍(ni-i+1)×Hn1,…,ni-1,ni-i+1,ni+1,…,nr(α)(x1,…,xr).
for ni≥i-1,    i=1,2,…,r; nj≥0, j≠i.Remark 27.
From Theorem11, the multivariable polynomials Hn1,…,nr(α)(x1,…,xr) have the following addition formula:
(81)Hn1,…,nr(α+β)(x1,…,xr)=∑k1=0n1‍⋯∑kr=0nr‍Hn1-k1,…,nr-kr(α)(x1,…,xr)×Hk1,…,kr(β)(x1,…,xr).Furthermore, settingmi=i, xi=0, yi=-xi, i=1,2,…,r in Corollary 5, we obtain the following class of bilinear generating functions for the polynomials Hn1,…,nr(α)(x1,…,xr).Remark 28.
If(82)Ξλ,η,α,β,γn,p(x1,…,xr;u1,…,ur;w):=∑k1=0[n1/p1]‍⋯∑kr=0[nr/pr]‍ak1,…,krHn1-p1k1,…,nr-prkr(α+β)(x1,…,xr)×Hλ1+η1k1,…,λr+ηrkr(γ)(u1,…,ur)w1k1⋯wrkr,
where ak1,…,kr≠0; pi∈ℕ; ni,λi,ηi∈ℕ0, i=1,2,…,r, and λ=(λ1,…,λr), η=(η1,…,ηr),w=(w1,…,wr), then we have
(83)∑l1=0n1‍⋯∑lr=0nr‍∑k1=0[l1/p1]‍⋯∑kr=0[lr/pr]‍ak1,…,krHn1-l1,…,nr-lr(α)(x1,…,xr)×Hl1-p1k1,…,lr-prkr(β)(x1,…,xr)×Hλ1+η1k1,…,λr+ηrkr(γ)(u1,…,ur)×w1k1⋯wrkr=Ξλ,η,α,β,γn,p(x1,…,xr;u1,…,ur;w).


## 5.2.1. The Case ofmi=1, i=1,2,…,r in Section 5.2

In this case, we get ar-variable analogue of Lagrange polynomials (see [14]) which is different from that in [10]
(64)(1-x1t1-⋯-xrtr)-α=∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)t1n1⋯trnr.They are given explicitly by(65)Gn1,…,nr(α)(x1,…,xr)=(α)n1+⋯+nrn1!⋯nr!x1n1⋯xrnr.Remark 21.
From Corollaries8 and 9, the multivariable polynomials Gn1,…,nr(α)(x1,…,xr) satisfy
(66)niGn1,…,nr(α)(x1,…,xr)=xi∂∂xiGn1,…,nr(α)(x1,…,xr)
for i=1,2,…,r and
(67)∑i=1r‍niGn1,…,nr(α)(x1,…,xr)=∑i=1r‍xi∂∂xiGn1,…,nr(α)(x1,…,xr).Remark 22.
By Theorem11, the multivariable polynomials Gn1,…,nr(α)(x1,…,xr) have the following addition formula:
(68)Gn1,…,nr(α+β)(x1,…,xr)=∑k1=0n1‍⋯∑kr=0nr‍Gn1-k1,…,nr-kr(α)(x1,…,xr)×Gk1,…,kr(β)(x1,…,xr).Remark 23.
From Theorem12, expansions of Gn1,…,nr(α)(x1,…,xr) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are given by
(69)Gn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑si=0[ni/2]‍(2ni-4si+1)si!(3/2)ni-siPni-2si(xi2)}×(α)n1+⋯+nr,Gn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑si=0[ni/2]‍(νi+ni-2si)si!(νi)ni-si+1Cni-2siνi(xi2)}×(α)n1+⋯+nr,Gn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑si=0[ni/2]‍Hni-2si(xi/2)si!(ni-2si)!}(α)n1+⋯+nr,Gn1,…,nr(α)(x1,…,xr)=∏i=1r‍{∑si=0ni‍(βi+1)ni2ni(-1)si(ni-si)!(βi+1)siLsi(βi)(xi2)}×(α)n1+⋯+nr.We can discuss some generating functions for the multivariable polynomialsGn1,…,nr(α)(x1,…,xr).Theorem 24.
For the polynomialsGn1,…,nr(α)(x1,…,xr),   the following generating function holds true for k∈ℕ0:
(70)∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)[k+ni]mz1n1⋯zrnr=m!(1-x1z1-⋯-xrzr)-α×gm(α,-k)(xizi1-x1z1-⋯-xrzr,-1),
for each i=1,2,…,r, where
(71)[k]m:=k(k-1)⋯(k-m+1),m∈ℕ,[k]0=1
and gn(α,β)(x,y) denotes the classical Lagrange polynomials defined by [14]
(72)(1-xt)-α(1-yt)-β=∑n=0∞‍gn(α,β)(x,y)tn(α,β∈ℂ;|t|<min{|x|-1,|y|-1}).Proof.
Fixi=1,2,…,r. Let 𝒞ξiand 𝒞ξi* denote the circles of radius ɛi, centered ξi=zi and ξi=0, respectively, where
(73)0<ɛi<|xi|-1|1-∑k=1r‍xkzk|.
Here these circles are described in the positive direction. By Cauchy's Integral Formula, we have
(74)Dzim{(1-x1z1-⋯-xrzr)-αzik}=m!2πi∮𝒞ξiξik(1-x1z1-⋯-xiξi-⋯-xrzr)-α(ξi-zi)m+1dξi=m!2πi×∮𝒞ξi*(ξi+zi)k(1-x1z1-⋯-xi(ξi+zi)-⋯-xrzr)-αξim+1dξi=m!2πizik(1-x1z1-⋯-xrzr)-α×∮𝒞ξi*(1+ξi/zi)kξim+1(1-xiξi1-x1z1-⋯-xrzr)-αdξi=m!zik(1-x1z1-⋯-xrzr)-α×gm(α,-k)(xi1-x1z1-⋯-xrzr,-1zi)=m!zik-m(1-x1z1-⋯-xrzr)-α×gm(α,-k)(xizi1-x1z1-⋯-xrzr,-1).
On the other hand, we can write that
(75)Dzim{(1-x1z1-⋯-xrzr)-αzik}=Dzim{zik∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)z1n1⋯zrnr}=∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)dmdzimz1n1⋯zini+k⋯zrnr=zik-m∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)[k+ni]mz1n1⋯zrnr.
From (74) and (75), we see that
(76)∑n1,…,nr=0∞‍Gn1,…,nr(α)(x1,…,xr)[k+ni]mz1n1⋯zrnr=m!(1-x1z1-⋯-xrzr)-α×gm(α,-k)(xizi1-x1z1-⋯-xrzr,-1).Corollary 25.
From (71), one observes that
(77)[ni+m]m=(ni+m)(ni+m-1)⋯(ni+1)=(ni+1)m
for each i=1,2,…,r. By setting k=min (70) and then using (77), one has
(78)∑n1,…,nr=0∞‍(ni+1)mGn1,…,nr(α)(x1,…,xr)z1n1⋯zrnr=m!(1-x1z1-⋯-xrzr)-α×gm(α,-m)(xizi1-x1z1-⋯-xrzr,-1).


## 5.2.2. The Case ofmi=i,i=1,2,…,r in Section 5.2

This case reduces to a multivariable analogue of Lagrange-Hermite polynomials(79)(1-x1t1-x2t22-⋯-xrtrr)-α=∑n1,…,nr=0∞‍Hn1,…,nr(α)(x1,…,xr)t1n1⋯trnr,
which is different from that in [6].Remark 26.
By Corollaries8 and 9, the multivariable polynomials Hn1,…,nr(α)(x1,…,xr) satisfy
(80)ixi∂∂xiHn1,…,ni-1,ni-i+1,ni+1,…,nr(α)(x1,…,xr)=(ni-i+1)Hn1,…,ni-1,ni-i+1,ni+1,…,nr(α)(x1,…,xr),∑i=1r‍ixi∂∂xiHn1,…,ni-1,ni-i+1,ni+1,…,nr(α)(x1,…,xr)=∑i=1r‍(ni-i+1)×Hn1,…,ni-1,ni-i+1,ni+1,…,nr(α)(x1,…,xr).
for ni≥i-1,    i=1,2,…,r; nj≥0, j≠i.Remark 27.
From Theorem11, the multivariable polynomials Hn1,…,nr(α)(x1,…,xr) have the following addition formula:
(81)Hn1,…,nr(α+β)(x1,…,xr)=∑k1=0n1‍⋯∑kr=0nr‍Hn1-k1,…,nr-kr(α)(x1,…,xr)×Hk1,…,kr(β)(x1,…,xr).Furthermore, settingmi=i, xi=0, yi=-xi, i=1,2,…,r in Corollary 5, we obtain the following class of bilinear generating functions for the polynomials Hn1,…,nr(α)(x1,…,xr).Remark 28.
If(82)Ξλ,η,α,β,γn,p(x1,…,xr;u1,…,ur;w):=∑k1=0[n1/p1]‍⋯∑kr=0[nr/pr]‍ak1,…,krHn1-p1k1,…,nr-prkr(α+β)(x1,…,xr)×Hλ1+η1k1,…,λr+ηrkr(γ)(u1,…,ur)w1k1⋯wrkr,
where ak1,…,kr≠0; pi∈ℕ; ni,λi,ηi∈ℕ0, i=1,2,…,r, and λ=(λ1,…,λr), η=(η1,…,ηr),w=(w1,…,wr), then we have
(83)∑l1=0n1‍⋯∑lr=0nr‍∑k1=0[l1/p1]‍⋯∑kr=0[lr/pr]‍ak1,…,krHn1-l1,…,nr-lr(α)(x1,…,xr)×Hl1-p1k1,…,lr-prkr(β)(x1,…,xr)×Hλ1+η1k1,…,λr+ηrkr(γ)(u1,…,ur)×w1k1⋯wrkr=Ξλ,η,α,β,γn,p(x1,…,xr;u1,…,ur;w).



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