Some generating functions involving certain

MATEMATIKA, 2017, Volume 33, Number 1, 71–85 c Penerbit UTM Press. All rights reserved

Some generating functions involving certain polynomials of two variables 1

Ahmed Ali Al-Gonah and 2 Hussein Abdulhafed Saleh

1,2 Department

of Mathematics, Aden University, Aden, Yemen e-mail: 1 [email protected]

Abstract In the present paper, we derive families of bilateral and mixed multilateral generating relations involving two variables Hermite and two variables extended Srivastava polynomials. Further, as applications of the main results, several bilateral and trilateral generating functions involving certain polynomials of two variables are obtained by using some recent known generating functions. Keywords Generating functions; Hermite polynomials; extended Srivastava polynomials. 2010 Mathematics Subject Classification 33B10, 33C45, 33E30.

1

Introduction ∞

Let {An,k }n,k=0 be a bounded double sequence of real or complex numbers. Over four decades ago, Srivastava [1] introduced the Srivastava polynomials (SP) SnN (x) by the following series definition: SnN (x) =

[ Nn ] X (−n)Nk k!

k=0

An,k xk (n ∈ N0 = N ∪ {0}; N ∈ N),

(1.1)

where N is the set of positive integer [a] denotes the greatest integer in a ∈ R and (λ)v , (λ)0 ≡ 1, denotes the Pochhammer symbol defined in terms of familiar Gamma function by [2, p. 22] Γ (λ + v) 1 (v = 0; λ ∈ C\ {0}) (λ)v = = . (1.2) λ (λ + 1) (λ + 2) · · · (λ + n − 1) (v = n ∈ N; λ ∈ C) Γ (λ) The Srivastava polynomials SnN (x) in (1.1) and their such interesting variants as follows: N Sn,q

(x) =

[ Nn ] X (−n)

Nk

k=0

k!

An+q,k xk (q, n ∈ N0 ; N ∈ N) ,

(1.3)

were investigated rather extensively in [3]. See [4,5] for further results. Clearly we have N Sn,0 (x) = SnN (x) .

(1.4)

Motivated essentially by the definitions (1.2) and (1.3), the following family of the two variables extended Srivastava polynomials (2VESP)Snq,N (x, y) was introduced and studied by Altın et al. [6]: Snq,N (x, y) =

[ Nn ] X

k=0

Aq+n,k

xn−Nk yk (q, n ∈ N0 ; N ∈ N) (n − N k)!k!

(1.5)

72

Ahmed Ali Al-Gonah and Hussein Abdulhafed Saleh

and further studied in [7]. However, in view of definitions (1.3) and (1.5), it is easily observed that ! n x y SN Snq,N (x, y) = n! n,q (−x)N which for x = −1 we get

(1.6)

n

Snq,N (−1, y) =

(−1) N S (y) . (n) ! n,q

(1.7)

It has been shown in [6] that the Snq,N (x, y) include many well-known polynomials of two variables such as Lagrange-Hermite polynomials, Lagrange polynomials and two-variables Hermite polynomials, under the special choices for the double sequence {An,k }. Here, we will recall them and add further new particular cases as the following remarks: Remark 1.1 ([6, p. 316] see also [7, p. 1116]). Choosing N = 1, Aq,n = (α)q−n (β)n (q, n ∈ N0 ) in equation (1.5), we get Snq,1 (x, y) = (α)q gn(α+q,β) (x, y) , (α,β)

where gn

(1.8)

(x, y) denotes the two variable Lagrange polynomials defined by [2 p. 441] gn(α,β) (x, y) =

n X (α)n−k (β)k

(n − k)!k!

k=0

xn−k yk

(1.9)

and specified by the following generating function (1 − xt)−α (1 − yt)−β =

∞ X

gn(α,β) (x, y) tn .

(1.10)

n=0

Remark 1.2 ([6, p. 316] see also [7, p. 1116]). Choosing N = 2, Aq,n = (α)q−2n (β)n (q, n ∈ N0 ) in equation(1.5), we get Snq,2 (x, y) = (α)q hn(α+q,β) (x, y) , (α,β)

where hn

(1.11)

(x, y) denotes the Lagrang–Hermite polynomials defined by [8] hn(α,β) (x, y)

=

[ n2 ] X (α)n−2k (β)k k=0

(n − 2k)!k!

xn−2k yk

(1.12)

and specified by the following generating function: −α

(1 − xt)

1 − yt

2 −β

=

∞ X

n=0

hn(α,β) (x, y) tn .

(1.13)

Some generating functions involving certain polynomials of two variables

73

Now, we add the following new particular cases as remarks: Remark 1.3 Choosing Aq,n = 1 (q, n ∈ N0 ) in equation (1.5), we get Snq,N (x, y) = (N)

where Hn

1 (N) H (x, y) , n! n

(1.14)

(x, y) denotes the Gould Hopper polynomials (GHP) defined by [9] Hn(N)

[ Nn ] X xn−Nk yk (x, y) = n! (n − N k)!k!

(1.15)

k=0

and specified by the following generating function: exp xt + yt

N

=

∞ X

Hn(N) (x, y)

n=0

tn . n!

(1.16)

Note that, for N = 1 in equation (1.14) and using the relation Hn(1) (x, y) = (x + y) we get

n

(1.17)

n

(x + y) . n! Also, for N = 2 in equation (1.14) and using the relation Snq,1 (x, y) =

(1.18)

Hn(2) (x, y) = Hn (x, y) ,

(1.19)

we get the known relation [6, p. 316] 1 H (x, y) , n!

Snq,2 (x, y) =

(1.20)

where Hn (x, y) denotes the two variables Hermit–Kamp de Friet polynomials (2VHKdFP) defined by [10, p. 341] [ n2 ] X xn−2k yk Hn (x, y) = n! (1.21) (n − 2k)!k! k=0

and specified by the following generating function: exp xt + yt

2

=

∞ X

n=0

Hn (x, y)

tn . n!

(1.22)

Remark 1.4 Choosing Aq,n = 1/n!, (q, n ∈ N0 ) in equation (1.5), we get Snq,N (y, x) =

1 N Ln (x, y) , n!

(1.23)

74


where N Ln (x, y) denotes the two variables generalized Laguerre polynomial (2VgLP) defined by [11 p. 213] [ Nn ] X xk yn−Nk (1.24) N Ln (x, y) = n! 2 (n − N k)! (k!) k=0 and specified by the following generating function

∞ X tn , exp (yt) C0 −xtN = N Ln (x, y) n! n=0

(1.25)

where C0 (x) denotes the 0th order Tricomi function. The nth order Tricomi functions Cn (x) are defined by [2] ∞ X (−1)k xk Cn (x) = . (1.26) k! (n + k)! k=0

For N = 1 and x = −x in equation (1.25), the 2VgLP N Ln (x, y) reduce to the two variables Laguerre polynomials (2VLP)Ln (x, y), i.e., we have 1 Ln (−x, y) =

Ln (x, y) ,

(1.27)

where Ln (x, y) defined by [11]: Ln (xy) = n!

n k X (−1) xk yn−k k=0

(n − k)! (k!)2

(1.28)

an specified by the following generating function exp (yt) C0 (−xt) =

∞ X

Ln (x, y)

n=0

tn . n!

(1.29)

Note that for N = 1 in equation (1.23) and using relation (1.27), we get Snq,1 (y, x) =

1 Ln (−x, y) . n!

(1.30)

Further for N = 2 in equation (1.23), we get Snq,2 (y, x) =

1 1 Sn (−x, y) , 2 Ln (x, y) = n! n!

(1.31)

where Sn (x, y) denotes the second form of the two variables Legendre polynomials (2VLeP) defined by [12] [ n2 ] k X (−1) xk yn−2k Sn (x, y) = n! (1.32) 2 (n − 2k)! (k!) k=0 an specified by the following generating function

∞ X tn exp (yt) C0 xt2 = Sn (x, y) . n! n=0

(1.33)

75

Some generating functions involving certain polynomials of two variables q−n

Remark 1.5 Choosing N = 1, Aq,n =

(−1) , (q, n ∈ N0 ) in equation (1.5), we get n! (q − n)!

Sn0,1 (x, y) =

Rn (x, y) (n!)2

,

(1.34)

where Rn (x, y) denotes the first form of the two variables Legendre polynomials defined by [13] n n−k n−k k X (−1) x y 2 Rn (x, y) = (n!) (1.35) 2 2. ((n − k)!) (k!) k=0

The aim of this paper is to derive some families of bilateral and mixed multilateral generating relations involving the 2VHKdFP Hn (x, y) and the 2VESP Snq,N (x, y) by using series rearrangement techniques. Also, the above mentioned remarks will be used to obtain some illustrative bilateral and trilateral generating functions involving the 2VHKdFP H n (x, y) and many other known two variables polynomials. For this aim we recall the following generating function for the 2VHKdFP Hn (x, y) [2] ! ∞ X tn α α+1 4yt2 −a , ; −; (α)n Hn (x, y) = (1 − xt) 2 F0 (1.36) 2 . n! 2 2 (1 − xt) n=0 The 2VHKdFP H n (x, y) are linked to the classical Hermite polynomials Hn (x) by the relations ix n Hn (x, y) = (−i) yn/2 Hn (1.37) √ 2 y and Hn (2x, −1) = Hn (x) ,

(1.38)

where the classical Hermite polynomials Hn (x) defined by [14, p. 187] Hn (x) = n!

[ n2 ] k n−2k X (−1) (2x) k=0

k! (n − 2k)!

Also, the 2VHKdFP H n (xy) satisfy the following relation: tn Hn (x, y) = Hn xt, yt2 .

2

.

(1.39)

(1.40)

Bilateral generating relations

We prove the following results: Theorem 2.1 The following family of bilateral generating relation involving the 2VHKdFP Hn (x, y) and the 2VESP Snq,N (x, y) holds true: n ∞ ∞ q X X ytN tn w q (w + xt) q,N Hq+n (u, v)Sn (x, y) = Hq+Nn (u, v)Aq+Nn,n . (2.1) q! n! q! q,n=0 n,q=0

76


Proof. Using definition (1.5), we have ∞ X

l.h.s =

Hq+n (u, v)

q,n=0

[ Nn ] X

Aq+n,k

k=0

xn−Nk yk tn w q (n − N k)!k! q!

(2.2)

which on replacing n by n + N k in the r.h.s. and using the lemma [2, p. 101] n [m ] ∞ X X

A (k, n) =

n=0 k=0

∞ X

A (k, n + mk) .

(2.3)

n=0 k=0

in the resultant equation becomes l.h.s =

∞ X ∞ X

Hn+q+Nk (u, v)An+q+Nk,k

n,q,k=0

xn yk tn+Nk w q . n! k! q!

(2.4)

Again replacing q by q − n in the r.h.s. of equation (2.4) and using the lemma [2, p. 100] ∞ X ∞ X

A (k, n) =

n=0 k=0

in the resultant equation, we get l.h.s =

∞ X

∞ X n X

n=0 k=0

Hq+Nk (u, v)Aq+Nk,k

q,k=0

A (k, n − k)

k q ytN X q!(xt)n w q−n . q!k! n=0 n!(q − n)!

(2.5)

(2.6)

Finally using the binomial theorem [2] (x + y)n =

n X n!xn− k yk k! (n − k)!

(2.7)

k=0

in the r.h.s. of the above equation, we get the r.h.s. of equation (2.1). 2 Remark 2.1. Taking w = −xt in assertion (2.1) of Theorem 2.1, we deduce the following consequence of Theorem 2.1: Corollary 2.1 The following family of bilateral generating relation involving the 2VHKdFP Hn (x, y) and the 2VESP Snq,N (x, y) holds true: n ∞ ∞ q X X ytN (−x) tn+q q,N Hq+n (u, v)Sn (x, y) = HNn (u, v)ANn,n (2.8) q! n! q,n=0 n=0 Remark 2.2 Taking w = 0 in assertion (2.1) of Theorem 2.1 we deduce the following consequence of Theorem 2.1: Corollary 2.2 The following family of bilateral generating relation involving the 2VHKdFP Hn (x, y) and the 2VESP Snq,N (x, y) holds true: n ∞ ∞ X X ytN (xt)q 0,N n Hn (u, v)Sn (x, y) t = Hq+Nn (u, v)Aq+Nn,n . (2.9) n! q! n=0 q,n=0 In the next section, Corollaries 2.1 and 2.2 will be exploited to derive families of mixed multilateral generating relations involving the 2VHKdFP Hn (x, y) and 2VESP Snq,N (x, y) with the help of the method considered in [2].

77


3

Multilateral generating relations

We prove the following theorem by using Corollary 2.1: Theorem 3.1 Corresponding to an identically non-vanishing function Ωµ (z1 , . . . , zl ) of complex variablesz1 , . . . , zl (l ∈ N) and of complex order µ , let Λµ,ψ (z1 , . . . , zl ; η) :=

∞ X

k=0

ak Ωµ+ψk (z1 , . . . , zl ) η k (ak 6= 0, ψ, η ∈ C) ,

(3.1)

then we have, for n, p ∈ N [ pq ] ∞ X ∞ X X

q−pk

ak Hq+n−pk (u, v)Snq−pk,N (x, y) Ωµ+ψk (z1 , . . . , zl ) η k

(−x)

q=0 n=0 k=0

= Λµ,ψ (z1 , . . . , z; η)

∞ X

HNn (u, v) ANn,n

n=0

q+n−pk

(wt) (q − pk)! n N y (wt) n!

, (3.2)

provided that each member of assertion (3.2) exists. Proof Using relation (2.3), we find l.h.s =

∞ X

ak Ωµ+ψk (z1 , . . . , zl ) η k

∞ X

q

Hq+n (u, v)Snq,N (x, y)

q,n=0

k=0

q+n

(−x) (wt) q!

(3.3)

Using equations (3.1) and (2.8) in the r.h.s. of equation (3.3), we get the r.h.s. of equation (3.2). 2 Next, following the same procedure leading to assertion (3.2) of Theorem 3.1 and using Corollary 2.2, we get the following result: Theorem 3.2 Corresponding to an identically non-vanishing function Ωµ (z1 , . . . , zl ) of complex variables z1 , . . . , zl (l∈ N) and of complex order µ, let Λµ,ψ (z1 , . . . , zl ; η):=

∞ X

k=0

Θµ,ψ n,p (u, v, x, y; z1 , . . . , zl ; τ ):=

[ np ] X

ak Ωµ+ψk (z1 , . . . , zl ) η k (ak 6= 0, ψ, η ∈ C) (3.4) 0,N ak H n−pk (u, v) Sn−pk (x, y)Ωµ+ψk (z1 , . . . , zl ) τ k

k=0

where n,p ∈ N. Then, we have ∞ X

n=0

Θµ,ψ n,p (u, v,x, y, z1 , . . . , zl ;

η n )t tp

=Λµ,ψ (z1 , . . . , zl ; η)

∞ X

Hq+Nn (u, v) Aq+Nn,n

q,n=0

provided that each member of assertion (3.5) exists.

ytN n!

n

q

(xt) . q!

(3.5)

78


Notice that, for every suitable choice of the coefficients ak (k ∈ N0 ), if the multivariable function Ωµ+ψk (z1 , . . . , zl ), (l ∈ N), is expressed in terms of simpler function of one and more variables, the assertions of Theorems 3.1 and 3.2 can be applied in order to derive various families of multilinear and multilateral generating relations involving some polynomials of two variables For example, if we set l = 2, ψ = 1, = 0, Ωk (z1 , z2 ) =Lk (z1 , z2 ) and ak = 1/k! in assertion (3.2) of Theorem 3.1 and using (1.28) we readily obtain the following mixed trilateral generating function: [ pq ] ∞ X ∞ X X

q−pk

q+n−pk

η k (−x) (wt) k! (q − pk)! q=0 n=0 k=0 n N ∞ y (wt) X . = exp (z2 η) C0 (−z1 η) HNn (u, v)ANn,n n! n=0 Hq+n−pk (u, v)Snq−pk,N (x, y) Lk (z1 z2 )

(3.6)

In the next section, we derive some bilateral and trilateral generating functions for the 2VHKdFP Hn (x, y) and other polynomials as applications of the results derived in Sections 2 and 3 with the help of certain generating functions and the remarks introduced in Section 1.

4

Applications

First, the following bilateral generating functions are obtained as applications of Corollaries 2.1 and 2.2: ∞ (i) Taking N = 1 and {Aq,n }q,n=0 as in Remark 1.1 and using relation (1.8) in equation (2.8), we get ∞ X

(α)q Hq+n (u, v) gn(α+q,β) (x, y)

q,n=0

∞ q n X (−x) tn+q (yt) = (β)n H (u, v) q! n! n=0

(4.1)

which on using equation (1.36) in the r.h.s. gives the following bilateral generating function: ∞ X

q

(α)q Hq+n (u, v)gn(α+q,β) (xy)

q,n=0

(−x) tn+q q!

= (1

− uyt)−β 2 F0

β β+1 4vy2 t2 , ; −; 2 2 2 (1 − uyt)

!

(α,β)

Now, using the following relation between the Jacobi polynomials Pn Lagrange polynomials [2, p. 442 ]: gn(α,β)

n

(x, y) = (y − x)

Pn(−α−n,−β−n)

x+y x−y

,

.

(4.2)

(x,y) and

(4.3)

79


in the r.h.s. of equation (4.2), we get ∞ X

(α)q Hq+n (u, v)Pn(−α−q−n,−β−n)

q,n=0

= (1 − uyt)

−β

2F0

(y − x)n (−x)q tn+q q! ! β β +1 4vy2 t2 , ; −; . 2 2 2 (1 − uyt)

x+y x−y

(4.4)

∞

Again taking N = 1 and {Aq,n }q,n=0 as in Remark 1.1 and using relation (1.8) in equation (2.9), we get ∞ X

Hn (u, v) gn(α,β) (x, y) tn =

n=0

∞ X

q,n=0

n

(α)q (β)n Hq+n (u, v)

q

(yt) (xt) n! q!

which on using relation (4.3) in the r.h.s. becomes ∞ X x+y n (yt − xt) Hn (u, v)Pn(−α−q−n,−β−n) x − y n=0 ∞ n q X (yt) (xt) = (α)q (β)n H q+n (u, v) . n! q! q,n=0

(4.5)

(4.6)

∞

(ii) Taking N = 2 and {Aq,n }q,n=0 as in Remark 1.2 and using relation (1.11) in equation (2.8), we get n ∞ ∞ q X X yt2 (−x) tn+q (α+q,β) (α)q Hq+n (u, v) hn (x, y) = (β)n H2n(u, v) . (4.7) q! n! q,n=0 n=0 ∞

Again taking N = 2 and {Aq,n }q,n=0 as in Remark 1.2 and using relation (1.11) in equation (2.9), we get n ∞ ∞ X X yt2 (xt)q (α,β) n Hn (u, v) hn (x, y) t = (α)n (β)n Hq+2n (u, v) . (4.8) n! q! q,n=0 n=0 ∞

(iii) Taking {Aq,n }q,n=0 as in Remark 1.3 and using relation (1.14) in equation (2.8), we get n ∞ ∞ q X X ytN (−x) tn+q Hq+n (u, v) Hn(N) (x, y) = HNn (u, v) , (4.9) n!q! n! q,n=0 n=0

which for N = 2 and then using relation (1.19) and the following generating function [15, p. 412]: X ∞ zn 1 u2 z √ exp = H2n (u, v) (4.10) 1 − 4vz n! 1 − 4vz n=0 in the l.h.s. and r.h.s. of the resultant equation respectively gives ∞ q X (−x) tn+q 1 u2 yt2 Hq+n (u, v) Hn (x, y) = p exp . n!q! 1 − 4vyt2 1 − 4vyt2 q,n=0

(4.11)

80

Ahmed Ali Al-Gonah and Hussein Abdulhafed Saleh ∞

Again taking {Aq,n }q,n=0 as in Remark 1.3 and using relation (1.14) in equation (2.9), we get n ∞ ∞ X X ytN (xt)q tn (N) Hn (u, v) Hn (x, y) = Hq+Nn (4.12) n! q,n=0 n! q! n=0 which for N = 2 and using relation (1.19) gives

n ∞ X yt2 (xt)q tn = Hq+2n (u, v) . Hn (u, v) Hn (x, y) n! q,n=0 n! q! n=0 ∞ X

(4.13)

Now, using the following bilinear generating function [16, p. 717 ]: tn 1 Hn (x, y) Hn (z, w) =p exp n! 1 − 4yt2 w n=0 X

xzt + t2 wx2 + yz 2 1 − 4yt2 w

!

(4.14)

in the l.h.s. of equation (4.13), we get 1 p

1 − 4yt2 v

exp

xut + t2 vx2 + yu2 1 − 4yt2 v

!

n yt2 (xt)q = Hq+2n (u, v) , (4.15) n! q! q,n=0 ∞ X

which for x = 0 and yt2 = z gives equation (4.10). ∞ (iv) Taking {Aq,n }q,n=0 as in Remark 1.4 and using relation (1.23) in equation (2.8), we get n ∞ ∞ q X X xtN (−y) tn+q = Hq+n (u, v) N Ln (x, y) HNn (u, v) (4.16) 2 . n!q! (n!) q,n=0 n=0 Now, for N = 1 and replacing x by −x in equation (4.16) and using equation (1.27) we get ∞ X

Hq+n (u, v) Ln (x, y)

q,n=0

n q X (−y) tn+q (−xt) = Hn (u, v) 2 , n!q! (n!) n=0

(4.17)

which on using relation (1.40) in the r.h.s. and then using the following definition of the nth order Hermite-Tricomi function [15, p. 407 ]: (for n = 0) H Cn (x, y)

=

∞ k X (−1) Hk (x, y) k=0

k! (k + n)!

(4.18)

gives the following bilateral generating function: ∞ X

q,n=0

q

Hq+n (u, v) Ln (x, y)

(−y) tn+q 2 = H C0 (uxt, vx2 t ). n!q!

(4.19)

Also for N = 2 and replacing x by −x in equation(4.16) and using equation (1.31), we get n ∞ ∞ q X X −xt2 (−y) tn+q Hq+n (u, v) Sn (x, y) = H2n (u, v) (4.20) 2 n!q! (n!) q,n=0 n=0


81

which on using relation (1.40) in the r.h.s. and then using the following definition of the 0th order Hermite-Bessel functions [12] H J0 (x, y)

=

∞ X (−1)r H2r (x, y) 2

22r (r!)

r=0

(4.21)

gives the following bilateral generating function: ∞ X

q

Hq+n (u, v) Sn (x, y)

q,n=0

√ (−y) tn+q = H J0 2u xt, 4vxt2 . n!q!

(4.22)

∞

Again taking {Aq,n }q,n=0 as in Remark 1.4 and using relation (1.23) in equation (2.9), we get n ∞ ∞ X X xtN (yt)q tn Hn (u, v) N Ln (x, y) = Hq+Nn (u, v) , (4.23) 2 n! q,n=0 q! (n!) n=0 which for N = 1 and replacing x by −x and using equation (1.27) becomes ∞ X

Hn (u, v) Ln (x, y)

n=0

∞ X tn (−xt)n (yt)q = . Hq+n (u, v) 2 n! q,n=0 q! (n!)

(4.24)

Now, using the following bilateral generating function [16, p. 718 ]: x tn = exp y (wt)2 + xwt H C0 2zt + ywt , y (zt)2 , n! 2 n=0 (4.25) in the l.h.s. of equation (4.24), we get u 2 2 + vyt , v (xt) exp v (yt) + uyt H C0 2xt 2 ∞ n q X (−xt) (yt) = Hq+n (u, v) . (4.26) 2 q! (n!) q,n=0 ∞ X

Hn (x, y) Ln (z, w)

Also for N = 2 in equation (4.23), we get ∞ X

n=0

Hn (u, v) Ln (x, y)

n ∞ X xt2 (yt)q tn = Hq+2n (u, v) , 2 n! q,n=0 q! (n!)

(4.27)

which on using the following bilateral generating function [16, p. 720]: ∞ X

tn Hn (x, y) 2 Ln (z, w) n! n=0 √ x 2 = exp xwt + y (wt) H J0 4y zt + wt , 4zyt2 , 2y

(4.28)

82


in the l.h.s. gives √ u 2 exp uyt + v (yt) H J0 4v xt + yt , 4xv2 2v n ∞ X xt2 (yt)q = Hq+2n (u, v) . 2 q! (n!) q,n=0

(4.29)

∞

(v) Taking N = 1 and {Aq,n }q,n=0 as in Remark 1.5 and using relation (1.34) in equation (2.9), we get ∞ X

Hn (u, v) Rn (x, y)

n=0

tn (n!)2

=

∞ X

Hq+n (u, v)

q,n=0

(yt)n (−xt)q (n!)2 (q!)2

.

(4.30)

Now using the following bilateral generating function [16, p. 721]: ∞ X

Hn (x, y)Rn (z, w)

n=0

tn 2

(n!)

2 2 = H C0,0 −xwt, y (wt) ; zxt, y (zt) | − 2wyzt2 .

(4.31)

in the l.h.s. of equation (4.30), we get

∞ n q X (yt) (−xt) 2 2 2 C −uyt, v (yt) ; xut, v (xt) − 2yvxt = H (u, v) (4.32) H 0,0 q+n 2 2 (n!) (q!) q,n=0

where H C0,0(α, β; γ, δ|ε) denotes the (0th , 0th ) order of the two-index Tricomi–Bessel function defined by [16] (for m = n = 0) H C0,0(x, y; z, w|τ )

∞ r+s X (−1) Hr,s (x, yz, w|τ ) 2

r,s=0

2

(r!) (s!)

.

(4.33)

Next, the following trilateral generating function is obtained as an application of the result (3.6): ∞ (vi) Taking N = 1 and {Aq,n }q,n=0 as in Remark 1.1 and using relation (1.8) in equation (3.6), we get [ pq ] ∞ X ∞ X X

q−pk

(α)q Hq+n−pk (u, v) gn(α+q−pk,β) (x, y) Lk (z1 z2 )

q=0 n=0 k=0

= exp (z2 η) C0 (−z1 η)

∞ X

q+n−pk

η k (−x) (wt) k! (q − pk)! n

(α)q H(u, v)

n=0

(ywt) , n!

(4.34)

which on using relation (1.36) in the r.h.s. give the following generating functions: [ pq ] ∞ X ∞ X X

q−pk

q=0 n=0 k=0

= exp (z2 η) C0 (−z1 η) (1 − uywt)−α 2 F0

q+n−pk

η k (−x) (wt) k! (q − pk)! ! 2 α α+1 4v (ywt) . (4.35) , ; −; 2 2 (1 − uywt)2

(α)q Hq+n−pk (u, v) gn(α+q−pk,β) (x, y) Lk (z1 z2 )

83


Similarly other trilateral generating functions can be obtained as applications of the result (3.6) with the help of Remarks 1.2–1.5 Finally, it is worthy to note that by using relation (1.38) the results obtained in this section give many bilateral and trilateral generating functions for the classical Hermite polynomials Hn (x) associated with other polynomials.

5

Concluding remarks

The approach adopted in this paper is general and can be exploited to establish further consequences regarding other special polynomials. Here, we establish some generating functions involving the 2VLP Ln (x, y) given in equations (1.28) and (1.29) Now, following the same procedure leading to assertion (2.1) of Theorem 2.1, we get the following family of bilateral generating relation involving the 2VLP Ln (x, y) and the 2VESP Snq,N (x, y): n ∞ ∞ X X ytN (w + xt)q tn w q q,N = Lq+Nn (u, v) Aq+Nn,n . (5.1) Lq+n (u, v) Sn (x, y) q! n! q! n,q=0 q,n=0 Putting w = −xt and w = 0 in equation (5.1) respectively, we get ∞ X

Lq+n (u, v)Snq,N

q,n=0

and

∞ X

Ln (u, v)Sn0,N

∞ q X ytN (−x) tn+q (x, y) = LNn (u, v)ANn,n q! n! n=0

n

(x, y) t =

n=0

∞ X

Lq+Nn (u, v)Aq+Nn,n

q,n=0

ytN n!

n

n

(5.2)

q

(xt) , q!

(5.3)

respectively. ∞ Taking {Aq,n }q,n=0 as in Remark 1.4 and using relation (1.23) in equation (5.2), we get n ∞ q X xtN (−y) tn+q Lq+n (u, v) N Ln (x, y) = LNn (u, v) 2 . n!q! (n!) q,n=0 n=0 ∞ X

(5.4)

For N = 1 and replacing x by −x in equation(5.4) and then using relation (1.27) and the identities tn Ln (x, y) = Ln (xt, yt) , (5.5) we get the following bilinear generating function: ∞ X

q

Lq+n (u, v) Ln (x, y)

q,n=0

(−y) tn+q = L C0 (uxt, vxy), n!q!

(5.6)

where L C0 (x, y) denotes the 0th order Laguerre-Tricomi functions defined by [12] L C0 (x, y)

=

∞ X (−1)k Lk (x, y) 2

k=0

(k!)

.

(5.7)

84

Ahmed Ali Al-Gonah and Hussein Abdulhafed Saleh ∞

Again taking {Aq,n }q,n=0 as in Remark 1.4 and using relation (1.23) in equation (5.3), we get n ∞ ∞ q X X xtN tn (yt) Ln (u, v) N Ln (x, y) = Lq+Nn (u, v) , (5.8) 2 n! q,n=0 q! (n!) n=0 which for N = 1 and replacing x by −x and using equation (1.27) becomes ∞ n q X tn (−xt) (yt) = Lq+n (u, v) . Ln (u, v) Ln (x, y) 2 n! q,n=0 q! (n!) n=0 ∞ X

(5.9)

Now, using the following bilinear generating function [17]: ∞ X

Ln (u, v) Ln (x, y)

n=0

tn = exp (vyt) C0,0 (xvt, yut, xut) , n!

(5.10)

in the l.h.s. of equation (5.9), we get exp (vyt) C0,0 (xvt, yut, xut) =

∞ X

q,n=0

n

Lq+n (u, v)

q

(−xt) (yt) , (n!)2 q!

(5.11)

where C0,0 (x, y, z) denotes the (0th , 0th ) order 2-indext 3-variable Tricomi functions defined by [18] ∞ X zk C0,0 (x, y, z) = Ck (x) Ck (y) . (5.12) k! k=0

Remark 5.1 As in Section 4 many generating functions can be obtained as applications of results (5.2 ) and (5.3) with the help of Remarks 1.1–1.3 and 1.5 Acknowledgements The authors would like to thank the referee for useful comments and suggestions.

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