Int Jr. of Mathematics Sciences & Applications Vol.3, No.1, January-June 2013 Copyright Mind Reader Publications ISSN No: 2230-9888 www.journalshub.com
Bilateral Generating Relations for Multi-Variable and Mixed-Type Special Polynomials Ghazala Yasmin* Department of Mathematics Aligarh Muslim University, Aligarh-202002, India Abstract: In the case of multi-variable special polynomials, the use of operational methods provides new means of analysis for the solutions of a wide class of partial differential equations often encountered in physical problems. In order to further stress the usefulness of operational methods, in this paper, we derive certain bilateral generating relations involving multi-variable and mixed-type special polynomials. Certain special cases are also discussed. Mathematics Subject Classification: 33C45; 33C47; 33E20. Keywords: Bilateral generating relations; Integral representations; multi-variable special polynomials. 1. Introduction and Preliminaries The 2-variable Hermite-Kamp´e de F´eriet polynomials (2VHKdFP) Hn (x, y) [2; p.341(23)] are defined as: ] [n 2 X xn−2r y r . (1.1) Hn (x, y) = n! (n − 2r)! r! r=0 In terms of the classical Hermite polynomials Hn (x) or Hen (x) [1], it is easily seen from definition (1.1) that Hn (2x, −1) = Hn (x) and
(1.2a)
1 Hn x, − = Hen (x). 2 Also, there exists the following close relationship [2; p.341(21)]: ix x n n/2 n n/2 Hn (x, y) = (−i) y Hn √ = i (2y) Hen √ , 2 y i 2y
(1.2b)
(1.3)
with the classical Hermite polynomials. The usage of a second variable (parameter) y in the 2VHKdFP Hn (x, y) is found to be convenient from the viewpoint of their applications. Also, we note that Hn (x, 0) = xn . *
(1.4)
Corresponding author; E-mail:
[email protected] (Ghazala Yasmin)
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We recall the diffusion equation, usually reffered as heat equation [3]: ∂ ∂2 F (x, y) = F (x, y), ∂y ∂x2
(1.5a)
F (x, 0) = g(x).
(1.5b)
The formal solution of the heat equation is given by ∂2 g(x). F (x, y) = exp y ∂x2
(1.6)
The presence of a second order derivative helps in getting an effective solution. The operator in the r.h.s. of equation (1.6) is usually reffered to as diffusive operator. For g(x) = xn , the natural solutions of heat equation are F (x, y) = Hn (x, y).
(1.7)
Thus 2VHKdFP Hn (x, y) are solutions of the heat equation and can therefore be introduced through an operational definition involving an evolution operator with a second order derivative as argument. Equations (1.4), (1.5) and (1.7) ensure that ∂ ∂2 Hn (x, y) = Hn (x, y), ∂y ∂x2
(1.8)
which in view of equation (1.4), gives the following operational definition for Hn (x, y): ∂2 {xn } . (1.9) Hn (x, y) = exp y ∂x2 Heat equation has many applications from engines to structural mechanics, the equation also has immense use in biology where it is known as diffusion equation and models the diffusion of a substance through a system. We recall some members of the family of Laguerre polynomials. (m)
The 3-variable associated Laguerre polynomials (3VaLP) Ln (x, y, z) are defined by the generating function [3; p.118(46)]: exp(zu + yt)C0 ((x − u)t) =
∞ X um tn (m) Ln (x, y, z), m! n! m,n=0
(1.10)
where C0 (x) denotes the 0th order Tricomi function. The nth order Tricomi function Cn (x) is defined by [1]: ∞ X (−x)r Cn (x) = . (1.11) r!(n + r)! r=0 2
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For y = z = 1 equation (1.10), reduces to the generating function for associated (m) Laguerre polynomials Ln (x) [3]: ∞ X um tn (m) Ln (x). m! n! m,n=0
exp(u + t)C0 ((x − u)t) =
(1.12) (ν)
Further, we consider the 2-variable associated Laguerre polynomials (2VaLP) ln (x, y) which are defined through the generating function [7; p.905(22)]: exp(yt)Cν (xt) =
∞ X tn n=0
n!
ln(ν) (x, y).
(1.13)
The pseudo Laguerre polynomials are defined by the generating function [4; p.79(2.7)]: j
r
exp(yt)x Cj (x t|r) =
∞ X tn n=0
where Cj (x|r) denotes the j
th
n!
Ln (x, y; r, j),
(1.14)
order Wright function defined by [8]: Cj (x|r) =
∞ X k=0
(−x)k . k!(j + kr)!
(1.15)
Further, we consider a mixed family of polynomials which provide a new point of view to the theory of generalizations of polynomials and functions. The Laguerre-based Hermite polynomials φn (x, y) are specified by the series [9; p.24(29)]: ] [n 2 X n!y r Ln−2r (x) . (1.16) φn (x, y) = r!(n − 2r)! r=0 The generating function for these polynomials is given by 2
exp(t + yt )C0 (xt) =
∞ X tn n=0
n!
φn (x, y),
(1.17)
Next, we consider the recently introduced Laguerre-based Appell polynomials L An (x, y) which are defined through the generating function [11; p.8(2.3a)]: A(t) exp(yt)C0 (xt) =
∞ X tn n=0
n!
L An (x, y).
(1.18)
We present the list of recently introduced Laguerre-based Appell polynomials in the following table: 3
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Table 1.1. List of Laguerre-based Appell polynomials S.No.
A(t)
t λ expt −1
α
II.
t λ expt −1
III.
t expt −1
α
IV.
t expt −1
V.
2 λ expt +1
α
VI.
2 λ expt +1
VII.
2 expt +1
α
VIII.
2 expt +1
I.
IX.
X. XI.
XII.
1 (1−t)m+1 1 1−t
1 (1−t)β
2t expt +1
Generating Functions α t exp(yt)C0 (xt) λ expt −1 P∞ n (α) t B (x, y; λ) n=0 n! L n t exp(yt)C0 (xt) t exp −1 Pλ∞ tn B (x, y; λ) n=0 n!Lα n t exp(yt)C0 (xt) expt −1 P∞ tn B (α) (x, y) L n=0 n! n t exp(yt)C0 (xt) expt −1 P∞ tn B (x, y) L n=0 n! n α 2 exp(yt)C0 (xt) λ expt +1 P∞ n (α) t E (x, y; λ) n=0 n! L n 2 exp(yt)C0 (xt) t exp +1 Pλ∞ tn E (x, y; λ) n=0 n!L n α 2 exp(yt)C0 (xt) expt +1 P∞ tn E (α) (x, y) L n=0 n! n 2 exp(yt)C0 (xt) expt +1 P∞ tn E (x, y) L n=0 n! n 1 exp(yt)C0 (xt) (1−t)m+1 P∞ (m) n G t (x, y) n L n=0 1 exp(yt)C0 (xt) 1−t P∞ n n=0 t L en (x, y) 1 exp(yt)C0 (xt) β (1−t) P∞ (β) n L fn (x, y) n=0 t 2t exp(yt)C 0 (xt) expt +1 P∞ ( n n=0 t L Gn x, y)
Polynomials Laguerre-Apostol-Bernoulli (α)
polynomials L Bn (x, y; λ) of order α Laguerre-Apostol-Bernoulli polynomials L Bn (x, y; λ) Laguerre-generalized Bernoulli (α)
polynomials L Bn (x, y) Laguerre Bernoulli polynomials L Bn (x, y)
Laguerre-Apostol-Euler (α)
polynomials L En (x, y; λ) of order α Laguerre-Apostol-Euler polynomials L En (x, y; λ)
Laguerre-generalized Euler (α)
polynomials L En (x, y) Laguerre Euler polynomials L En (x, y)
Laguerre- Miller-Lee (m)
polynomials L Gn (x, y) Laguerre- truncated exponential polynomials L en (x, y) Laguerre- modified Laguerre (β)
polynomials L fn (x, y) Laguerre- Genocchi polynomials L Gn (x, y)
The operational techniques include integral, differential and exponential operators and provide a systematic and analytic approach in the study of special functions. To further stress the importance of operational methods, in this paper, we derive certain bilateral generating relations for multi-variable and mixed-type special polynomials. 2. Generating Relations for Multi-Variable Special Polynomials We prove the following bilateral generating relations for multi-variable special polynomials: Theorem 2.1. If there exist the following bilateral generating relation involving 2VHKdFP (m) Hn (x, y) and 3VaLP Ln (z, w, p): ∞ X um tn Hn (x, y)L(m) F (x, y, z, w, p|u; t) = n (z, w, p), m! n! m,n=0
(2.1)
then the following generating relation holds true: ∞ X um tn exp(pu+xwt+yw t )H Co ((x+2ywt)(z−u)t, y(z−u) t ) = Hn (x, y)L(m) n (z, w, p). m! n! m,n=0 2 2
2 2
(2.2) 4
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where H C0 (x, y) denotes the 0th order Hermite-Tricomi functions. The nth order HermiteTricomi functions H Cn (x, y) are defined by [6]: H Cn (x, y)
=
∞ X (−1)r Hr (x, y) r=0
r!(n + r)!
.
(2.3)
Proof. In view of the operational definition (1.9) and generating function (1.10), we have 2 ∞ X um tn ∂ Hn (x, y)L(m) exp y 2 {exp(pu + wxt)Co ((z − u)xt)} = n (z, w, p). ∂x m! n! m,n=0 Now, using the Gauss transformation formula [5]: 2 Z −∞ ∂ −(x − ξ)2 1 exp y 2 {f (x)} = √ exp f (ξ)dξ, ∂x 2 πy −∞ 4y
(2.4)
(2.5)
in the l.h.s. of equation (2.4), we find 2 Z −∞ x2 ξ x 1 exp − + wt + ξ Co ((z − u)ξt)dξ √ exp pu − 2 πy 4y 4y 2y −∞ ∞ X um tn Hn (x, y)L(m) = n (z, w, p), m! n! m,n=0
which on using the integral representation [10]: r 2 Z −∞ b π bc c2 2 exp , exp(−ax + bx)Cn (cx)dx = , H Cn a 4a 2a 4a −∞
(2.6)
(2.7)
gives the generating relation (2.2). Corollary 2.1. If there exist the following bilateral generating relation involving 2VHKdFP (m) Hn (x, y) and associated Laguerre polynomials Ln (z): ∞ X um t n F (x, y, z, |u; t) = Hn (x, y)L(m) n (z), m! n! m,n=0
(2.8)
then the following generating relation holds true: ∞ X um tn Hn (x, y)L(m) exp(u + xt + yt )H Co ((x + 2yt)(z − u)t, y(z − u) t ) = n (z). (2.9) m! n! m,n=0 2
2 2
Proof. Taking p = w = 1 in equation (2.2), we get the generating relation (2.9).
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Moreover, we have the following alternative proof of Corollary 2.1: In view of the operational definition (1.9) and generating function (1.12), we have ∂2 exp y 2 ∂x
∞ X um tn {exp(u + xt)Co ((z − u)xt)} = Hn (x, y)L(m) n (z). m! n! m,n=0
(2.10)
Now, using the Gauss transformation formula (2.5) in the l.h.s of equation (2.10), we find Z −∞ 2 1 x2 ξ x exp − + t + ξ Co ((z − u)ξt)dξ √ exp u − 2 πy 4y 4y 2y −∞ ∞ X um tn Hn (x, y)L(m) = n (z), m! n! m,n=0
(2.11)
which on using the integral representation (2.7), gives the bilateral generating relation (2.9). Remark 2.1. For m = p = 0 equation (2.2), gives the bilateral generating relation [5; p.719(17)] 2 2
2 2
exp(xwt + yw t )H Co ((x + 2ywt)zt, yz t ) =
∞ X tn n=0
n!
Hn (x, y)Ln (z, w).
(2.12)
Remark 2.2. For u = 0, equation (2.8), reduces to the bilateral generating relation 2
2 2
exp(xt + yt )H Co ((x + 2yt)zt, yz t ) =
∞ X tn n=0
n!
Hn (x, y)L(m) n (z).
(2.13)
Theorem 2.2. If there exist the following bilateral generating relation involving 2VHKdFP (ν) Hn (x, y) and 2VaLP ln (z, w): F (x, y, z, w|; t) =
∞ X tn n=0
n!
Hn (x, y)ln(ν) (z, w),
(2.14)
then the following generating relation holds true: 2 2
2 2
exp(xwt + yw t )H Cν ((x + 2ywt)zt, yz t ) =
∞ X tn n=0
n!
Hn (x, y)ln(ν) (z, w).
(2.15)
Proof. In view of the operational definition (1.9) and generating function (1.13), we have 6
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Bilateral Generating Relations for Multi-Variable…
2 ∞ X ∂ tn exp y 2 {exp(wxt)Cν (zxt)} = Hn (x, y)ln(ν) (z, w). ∂x n! n=0
(2.16)
Again using the Gauss transformation formula (2.5) in the l.h.s. of equation (2.16), we find 2 Z −∞ 2 ∞ X 1 −x x tn ξ ξ Cν (zξt)dξ = Hn (x, y)ln(ν) (z, w), exp − + wt + √ exp 2 πy 4y 4y 2y n! −∞ n=0 (2.17) which in accordance with the integral representation (2.7), gives the generating relation (2.15). Remark 2.3. For ν = 0, equation (2.15), gives the bilateral generating relation (2.12). Theorem 2.3. If there exist the following bilateral generating relation involving 2VHKdFP Hn (x, y) and the pseudo Laguerre polynomials Ln (z, w; r, j): F (x, y, z, w; t) =
∞ X tn n=0
n!
Hn (x, y)Ln (z, w; r, j),
(2.18)
then the following generating relation holds true: z j exp(xwt+yw2 t2 )H Cj ((x+2ywt)(z r t), yz 2r t2 |r) =
∞ X tn n=0
n!
Hn (x, y)Ln (z, w; r, j), (2.19)
Proof. In view of the operational definition (1.9) and generating function (1.14), we have 2 ∞ X ∂ tn exp y 2 {exp(xwt)z j Cj (z r xt|r)} = Hn (x, y)Ln (z, w; r, j). ∂x n! n=0
(2.20)
Now, using the Gauss transformation formula (2.5) in the l.h.s of equation (2.20), we find 1 √ exp 2 πy
−x2 4y
Z
−∞
exp
−∞
=
∞ X tn n=0
n!
−ξ 2 x ξ z j Cj (z r ξt|r)dξ + wt + 4y 2y
Hn (x, y)Ln (z, w; r, j),
(2.21)
which on using the integral representation (2.7), gives the generating relation (2.19).
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Remark 2.4. For j = 0 and r = 1, equation (2.19), reduces to the bilateral generating relation (2.12). 3. Generating Relations for Mixed Type Special Polynomials We prove the following bilateral generating relations involving mixed type special polynomials: Theorem 3.1. If there exist the following bilateral generating relation involving 2VHKdFP Hn (x, y) and Laguerre-based Hermite polynomials φn (z, w): F (x, y, z, w; t) =
∞ X tn n=0
n!
Hn (x, y)φn (z, w),
(3.1)
then the following generating relation holds true: 1
p exp 1 − 4ywt2
xt + yt2 1 − 4ywt2
H C0
(x + 2yt)zt yz 2 t2 , 1 − 4ywt2 1 − 4ywt2
=
∞ X tn n=0
n!
Hn (x, y)φn (z, w). (3.2)
Proof. In view of the operational definition (1.9) and generating function (1.17), we have ∂2 exp y 2 ∂x
2 2
{(exp(xt + wx t )Co (zxt)} =
∞ X tn n=0
n!
Hn (x, y)φn (z, w).
(3.3)
Now, using the Gauss transformation formula (2.5) in the l.h.s. of equation (3.3), we find 1 √ exp 2 πy
−x2 4y
Z
−∞
−∞
1 x 2 2 exp − − wt ξ + t + ξ Co (zξt)dξ 4y 2y
=
∞ X tn n=0
n!
Hn (x, y)φn (z, w),
(3.4)
which on using the integral representation (2.7), gives the generating relation (3.2). Theorem 3.2. If there exist the following bilateral generating relation involving 2VHKdFP Hn (x, y) and Laguerre-based Appell polynomials L An (z, w): F (x, y, z, w; t) =
∞ X tn
n!
n=0
8
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Hn (x, y)L An (z, w),
(3.5)
Bilateral Generating Relations for Multi-Variable…
then the following generating relation holds true: 2 2
2 2
A(t) exp(xwt + yw t )H Co ((x + 2ywt)zt, yz t ) =
∞ X tn n=0
n!
Hn (x, y)L An (z, w).
(3.6)
Proof. In view of the operational definition (1.9) and generating function (1.18), we have ∂2 exp y 2 ∂x
{A(t) exp(xwt)Co (xzt)} =
∞ X tn n=0
n!
Hn (x, y)L An (z, w).
(3.7)
Again, using the Gauss transformation formula (2.5) in the l.h.s. of equation (3.7), we find 2 Z −∞ 2 −x ξ x 1 exp − + wt + ξ Co (zξt)dξ √ A(t) exp 2 πy 4y 4y 2y −∞ =
∞ X tn n=0
n!
Hn (x, y)L An (z, w),
(3.8)
which on using the integral representation (2.7), gives the bilateral generating relation (3.6). Corollary 3.1. The following bilateral generating relation involving 2VHKdFP Hn (x, y) (α) and Laguerre Apostol Bernoulli polynomials L Bn (z, w; λ) of order α holds true:
t λ expt −1
α
2 2
2 2
exp(xwt+yw t )H Co ((x+2ywt)zt, yz t ) =
∞ X tn n=0
n!
Hn (x, y)L Bn(α) (z, w; λ). (3.9)
Proof. If A(t) =
t λ expt −1
α
, (Table 1.1.(I)), then L An (z, w) in r.h.s of equation (3.6) (α)
are the Laguerre Apostol Bernoulli polynomials L Bn (z, w; λ) of order α and thus equation (3.6) gives the bilateral generating relation (3.9). Remark 3.1. For α = 1, equation (3.9) gives the following bilateral generating relation for Laguerre Apostol Bernoulli polynomials L Bn (z, w; λ) (Table 1.1.(II)):
t λ expt −1
2 2
2 2
exp(xwt + yw t )H Co ((x + 2ywt)zt, yz t ) =
∞ X tn n=0
n!
Hn (x, y)L Bn (z, w; λ) (3.10)
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and for λ = 1, equation (3.9) gives the following bilateral generating relation for Laguerre (α) generalized Bernoulli polynomials L Bn (z, w) (Table 1.1.(III)): α ∞ X t tn 2 2 2 2 exp(xwt + yw t ) C ((x + 2ywt)zt, yz t ) = Hn (x, y)L Bn(α) (z, w). H o expt −1 n! n=0 (3.11) Further for α = 1, equation (3.11) gives the bilateral generating relation for Laguerre Bernoulli polynomials L Bn (z, w) (Table 1.1.(IV)). Corollary 3.2. The following bilateral generating relation involving 2VHKdFP Hn (x, y) (α) and Laguerre Apostol Euler polynomials L En (z, w; λ) of order α holds true: α ∞ X 2 tn 2 2 2 2 exp(xwt+yw t ) C ((x+2ywt)zt, yz t ) = Hn (x, y)L En(α) (z, w; λ). H o λ expt +1 n! n=0 (3.12) α Proof. If A(t) = λ exp2t +1 , (Table 1.1.(V)), then L An (z, w) in r.h.s of equation (3.6) (α)
are the Laguerre Apostol Euler polynomials L En (z, w; λ) of order α and thus equation (3.6) gives the bilateral generating relation (3.12). Remark 3.2. For α = 1, equation (3.12) gives the following bilateral generating relation for Laguerre Apostol Euler polynomials L En (z, w; λ) (Table 1.1.(VI)): ∞ X tn 2 2 2 2 2 Hn (x, y)L En (z, w; λ) exp(xwt + yw t ) C ((x + 2ywt)zt, yz t ) = H o λ expt +1 n! n=0 (3.13) and for λ = 1, equation (3.12) gives the following bilateral generating relation for Laguerre (α) generalized Euler polynomials L En (z, w) (Table 1.1.(VII)): α ∞ X 2 tn 2 2 2 2 exp(xwt + yw t ) C ((x + 2ywt)zt, yz t ) = Hn (x, y)L En(α) (z, w). H o t exp +1 n! n=0 (3.14) Also, for α = 1, equation (3.14) gives the bilateral generating relation for Laguerre Euler polynomials L En (z, w) (Table 1.1.(VIII)). Corollary 3.3. The following bilateral generating relation involving 2VHKdFP Hn (x, y) (m) and Laguerre-Miller-Lee polynomials L Gn (x, y) holds true: ∞ X 1 tn 2 2 2 2 exp(xwt + yw t )H Co ((x + 2ywt)zt, yz t ) = Hn (x, y)L G(m) n (x, y). m+1 (1 − t) n! n=0 (3.15) 10
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Proof. If A(t) =
1 (1−t)m+1
α
, (Table 1.1.(IX)), then L An (z, w) in r.h.s of equation (m)
(3.6) are the Laguerre-Miller-Lee polynomials L Gn (x, y) and thus equation (3.6), gives the bilateral generating relation (3.15). Remark 3.3. For m = β − 1, equation (3.15) gives the following bilateral generating relation for Laguerre-modified Laguerre polynomials L fn (β)(x, y) (Table 1.1.(XI)):
1 (1 − t)β
exp(xwt + yw2 t2 )H Co ((x + 2ywt)zt, yz 2 t2 ) =
∞ X tn n=0
n!
Hn (x, y)L fn(β−1) (x, y)
(3.16) and for m = 0, equation (3.15) gives the following bilateral generating relation for Laguerre-truncated exponential polynomials L en (x, y) (Table 1.1.(X)):
1 1−t
2 2
2 2
exp(xwt + yw t )H Co ((x + 2ywt)zt, yz t ) =
∞ X tn n=0
n!
Hn (x, y)L en (x, y). (3.17)
Corollary 3.4. The following bilateral generating relation involving 2VHKdFP Hn (x, y) and Laguerre- Genocchi polynomials L Gn (x, y) holds true:
2t expt + 1
2 2
2 2
exp(xwt + yw t )H Co ((x + 2ywt)zt, yz t ) =
∞ X tn n=0
n!
Hn (x, y)L Gn (x, y).
(3.18) 2t Proof. If A(t) = , (Table 1.1.(XII)), then L An (z, w) in r.h.s of equation (3.6) expt +1 are the Laguerre-Genocchi polynomials L Gn (x, y) and thus equation (3.6) gives the bilateral generating relation (3.18).
4. Concluding Remark The operational techniques (including differential and integral operators) provide a systematic and analytic approach in the study of special functions. In the previous sections, we have derived the bilateral generating relations for multi-variable and mixed-type special polynomials by using the gauss transformation formula and integral representations as basic tools. The method outlined in the previous section is general and allow us to derive bilateral generating relations involving more general forms of multi-variable special polynomials. (µ,ν) To give an example, first we recall the generalized Legendre polynomials Rn (x, y) which are defined as [7; p.907(38)]: Rn(µ,ν) (x, y)
2
= (n!)
n X r=0
(−1)n−r xn−r y r r!(n − r)!Γ(n − r + µ + 1)Γ(r + ν + 1)
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(4.1)
Ghazala Yasmin
and satisfy the generating function ∞ X tn (µ,ν) Cν (−yt)Cµ (xt) = R (x, y). 2 n (n!) n=0
(4.2)
∞ X √ √ tn (µ,ν) (−yt) (xt) Jν (2 −yt)Jµ (2 xt) = (x, y). R (n!)2 n n=0
(4.3)
Also, we note that ν 2
µ 2
Now, If we consider the following bilateral generating relation involving 2VHKdFP (µ,ν) Hn (x, y) and generalized Legendre Polynomials Rn (z, w): ∞ X tn (µ,ν) (z, w), F (x, y, z, w; t) = 2 Hn (x, y)Rn (n!) n=0
(4.4)
then the following generating relation holds true: ∞ X tn (µ,ν) (z, w), H Cµ,ν (−xwt, yw t ; zxt, yz t | − 2ywzt ) = 2 Hn (x, y)Rn (n!) n=0 2 2
where [5]:
H Cµ,ν
2 2
2
(4.5)
is the (µth , ν th ) order of the two index Hermite-Tricomi function defined as ∞ X (−1)r+s Hr,s (x, y; z, w|τ ) . C (x, y; z, w|τ ) = H m,n r!s!(m + s)!(n + r)! r,s=0
(4.6)
Now, using the operational definition (1.9) and generating function (4.2), the r.h.s. of the bilateral generating relation (4.4) can be expressed as 2 ∂ exp y 2 {Cµ (xzt)Cν (−xwt)}. (4.7) ∂x Applying the Gauss transformation formula (2.5) on equation (4.7), we get 2 2 Z −∞ ξ −x x 1 exp − + ξ Cµ (zξt)Cν (−wξt)dξ, √ exp 2 πy 4y 4y 2y −∞
(4.8)
which on using the integral representation [5]: r 2 Z −∞ π b bc c2 bd d2 dc 2 exp(−ax + bx)Cµ (cx)Cν (dx)dx = exp , , , | , H Cµ,ν a 4a 2a 4a 2a 4a 2a −∞ (4.9) gives the generating relation (4.5).
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Bilateral Generating Relations for Multi-Variable…
For µ = ν = 0, equation (4.5) gives the following bilateral generating relation [5; p.721(33)]: ∞ X tn Hn (x, y)Rn (z, w). H C0,0 (−xwt, yw t ; zxt, yz t | − 2ywzt ) = (n!)2 n=0 2 2
2 2
2
(4.10)
It is evident that the method used in this paper is efficient in providing families of bilateral generating relations. The extention of this method to obtain the bilateral generating relations for general families of polynomials and functions, is an interesting problem for further research. References [1] L.C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan Publishing Company, New York, 1985. [2] P. Appell and J. Kamp´e de F´ eriet, Fonctions Hyperg´eom´etriques et Hypersph´eriques: Polynˆomes d’ Hermite, Gauthier-Villars, Paris, 1926. [3] G. Dattoli, Generalized polynomials, operational identities and their applications, J. comput. Appl. Math. 118 (2000), pp.111-123. [4] G. Dattoli, Pseudo Laguerre and Pseudo Hermite polynomials, Rend. Mat. Acc. Linceri. 12(9) (2001), pp.75-84. [5] G. Dattoli, Bilateral generating functions and operational methods, J. Math. Anal. Appl. 269 (2002), pp.716-725. [6] G. Dattoli, M. Mancho and A. Torre, The generalized Laguerre polynomials, the associated Bessel functions and application to propagation problems, Radiat. Phys. Chem. 59 (2000), pp.229-237. [7] G. Dattoli and M. Migliorati, Associated Laguerre polynomials: Monomiality and bi-orthogonal functions, Int. Math. Forum 3(19) (2008), pp.901-909. [8] G. Dattoli and A. Torre, Theory and Applications of Generalized Bessel Functions, Aracne, Rome, 1996. [9] G. Dattoli and A. Torre, Exponential operator, quasi-monomials and generalized polynomials, Radiat.Phys.Chem.57 (2000), pp.21-26. [10] G. Dattoli and A. Torre, S. Lorenzutta and C. Cesarano, Generalized polynomials and operatorial identities, Acc. Sc. Torino-Atti Sc. Fis 134 (2000), pp.231-249. [11] Subuhi Khan, M,W,A. Saad and R. Khan, Laguerre-based Appell polynomials: Properties and applications, Math. Compu. Modelling 52 (2010), pp.247-259.
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