论文部分内容阅读
Bingbing Li , Ying Zhang
1.Anyang Vocational and Technical College , He Nan Province,455000,china
2.No.55 Middle School of Luoyang, He Nan Province, 471000 ,china
Abstract: By Mellin integral transforms for the functionmultiplied byof the type (with an arbitrary constant ,we obtain the Mellin integral transforms for the q-Krawtchouk polynomials, the q-Meixner polynomials.
Key Words: Mellin integral transforms; q-polynomials
1 Introduction
We recall some standard notations for q-shifted factorial and basic hypergeometric series in[1]. For two complex a and q, the shifted factorial with base q is defined by
for ,.
As usual, the basic hypergeometric series is defined by
,.
The Ramanujan's q-extension [2, 3] of the Euler representation for the gamma function is,(1)
Where , .
It was shown in [4] that by using a q-analogue of Euler's reflection formula
is the theta-function of Jacobi, one can represent (1) in the form
(2)
wherefor any nonnegative integer n. Mellin integral transform is defined by. It is easy to see that, when , Mellin integral transform reduces to the Euler representation for the gamma function , When , .
Using (2), we have .
In [5], ,(3)
Whereis a polynomial of degree n in x. When,the Mellin integral transform (3) turns to
,(4)
By, wehave , we have
(5)
Several examples
First, take q-Krawtchouk polynomialsfor example, we have the following Mellin integral transform:
Transformation 1:
(6)
Let (7) substituting (7) in to (4),we obtain the following Mellin integral transform for the q-Krawtchouk polynomials :
Transformation 2:
Similarly, for the q-Meixner polynomials ,we have
Transformation 3:
.
References
[1] G. Gasper, M. Rahman, Basic Hypergeometric Series, second ed., Cambridge University
Press, Cambridge, 2004.
[2] R. Askey, Ramanujans extensions of the gamma and beta functions, Amer. Math.
Monthly 87 (5) (1980) 346-359.
[3] R. Askey, Beta integrals and q-extensions, in: Proceedings of the Ramanujan Centen-
nial International Conference, Annamalainagar, December 15-18, 1987, The Ramanujan
Mathematical Society, Annamalainagar, India, 1988, pp. 85-102.
[5] N.M. Atakishiyev, M.K. Atakishiyeva, A q-analogue of the Euler gamma integral, The-
oret. Math. Phys. 129 (1) (2001) 1325-1334.
[6] S. Ramanujan, Some definite integrals, Messenger Math. 44 (1915) 10-18 (reprinted
in: G.H. Hardy, P.V. Seshu Aiyar, B.M. Wilson (Eds.), Collected Papers of Srinivasa
Ramanujan, Cambridge University Press, Cambridge, 1927, pp.53-58).
1.Anyang Vocational and Technical College , He Nan Province,455000,china
2.No.55 Middle School of Luoyang, He Nan Province, 471000 ,china
Abstract: By Mellin integral transforms for the functionmultiplied byof the type (with an arbitrary constant ,we obtain the Mellin integral transforms for the q-Krawtchouk polynomials, the q-Meixner polynomials.
Key Words: Mellin integral transforms; q-polynomials
1 Introduction
We recall some standard notations for q-shifted factorial and basic hypergeometric series in[1]. For two complex a and q, the shifted factorial with base q is defined by
for ,.
As usual, the basic hypergeometric series is defined by
,.
The Ramanujan's q-extension [2, 3] of the Euler representation for the gamma function is,(1)
Where , .
It was shown in [4] that by using a q-analogue of Euler's reflection formula
is the theta-function of Jacobi, one can represent (1) in the form
(2)
wherefor any nonnegative integer n. Mellin integral transform is defined by. It is easy to see that, when , Mellin integral transform reduces to the Euler representation for the gamma function , When , .
Using (2), we have .
In [5], ,(3)
Whereis a polynomial of degree n in x. When,the Mellin integral transform (3) turns to
,(4)
By, wehave , we have
(5)
Several examples
First, take q-Krawtchouk polynomialsfor example, we have the following Mellin integral transform:
Transformation 1:
(6)
Let (7) substituting (7) in to (4),we obtain the following Mellin integral transform for the q-Krawtchouk polynomials :
Transformation 2:
Similarly, for the q-Meixner polynomials ,we have
Transformation 3:
.
References
[1] G. Gasper, M. Rahman, Basic Hypergeometric Series, second ed., Cambridge University
Press, Cambridge, 2004.
[2] R. Askey, Ramanujans extensions of the gamma and beta functions, Amer. Math.
Monthly 87 (5) (1980) 346-359.
[3] R. Askey, Beta integrals and q-extensions, in: Proceedings of the Ramanujan Centen-
nial International Conference, Annamalainagar, December 15-18, 1987, The Ramanujan
Mathematical Society, Annamalainagar, India, 1988, pp. 85-102.
[5] N.M. Atakishiyev, M.K. Atakishiyeva, A q-analogue of the Euler gamma integral, The-
oret. Math. Phys. 129 (1) (2001) 1325-1334.
[6] S. Ramanujan, Some definite integrals, Messenger Math. 44 (1915) 10-18 (reprinted
in: G.H. Hardy, P.V. Seshu Aiyar, B.M. Wilson (Eds.), Collected Papers of Srinivasa
Ramanujan, Cambridge University Press, Cambridge, 1927, pp.53-58).