CERTAIN NEW GENERATING RELATIONS FOR PRODUCTS OF TWO LAGUERRE POLYNOMIALS

Title & Authors
CERTAIN NEW GENERATING RELATIONS FOR PRODUCTS OF TWO LAGUERRE POLYNOMIALS
CHOI, JUNESANG; RATHIE, ARJUN KUMAR;

Abstract
Generating functions play an important role in the investigation of various useful properties of the sequences which they generate. Exton [13] presented a very general double generating relation involving products of two Laguerre polynomials. Motivated essentially by Exton's derivation [13], the authors aim to show how one can obtain nineteen new generating relations associated with products of two Laguerre polynomials in the form of a single result. We also consider some interesting and potentially useful special cases of our main findings.
Keywords
gamma function;hypergeometric function;generalized hypergeometric function;$\small{Kamp{\acute{e}}}$ de $\small{F{\acute{e}}riet}$ function;Kummer's second summation theorem;Dixon and Whipple's summation theorems;
Language
English
Cited by
References
1.
P. Appell and J. Kampe de Feriet, Fonctions Hypergeometriques et Hyperspheriques; Polynomes d'Hermite, Gauthier-Villars, Paris, 1926.

2.
W. N. Bailey, Identities of the Rogers-Ramanujan type, Proc. London Math. Soc. 50 (1948), 1-10.

3.
W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935; Reprinted by Stechert Hafner, New York, 1964.

4.
Yu. A. Brychkov, Handbook of Special Functions, CRC Press, Taylor & Fancis Group, Boca Raton, London, New York, 2008.

5.
J. L. Burchnall and T. W. Chaundy, Expansions of Appell's double hypergeometric functions, Quart. J. Math. (Oxford Ser.) 11 (1940), 249-270.

6.
J. L. Burchnall and T. W. Chaundy, Expansions of Appell's double hypergeometric functions. II, Quart. J. Math. (Oxford Ser.) 12 (1941), 112-128.

7.
R. G. Buschman and H. M. Srivastava, Series identities and reducibility of Kampe de Feriet functions, Math. Proc. Cambridge Philos. Soc. 91 (1982), no. 3, 435-440.

8.
J. Choi, Notes on formal manipulations of double series, Commun. Korean Math. Soc. 18 (2003), no. 4, 781-789.

9.
J. Choi and A. K. Rathie, Reducibility of Kampe de Feriet function, Preprint, 2015.

10.
J. Choi and A. K. Rathie, Reducibility of certain Kampe de Feriet function with an application to generating relations for products of two Laguerre polynomials, Filomat, 2015, accepted for publicatiuon.

11.
A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions. Vol. I, McGraw-Hill Book Company, New York, Toronto and London, 1953.

12.
H. Exton, Multiple Hypergeometric Functions and Applications, Halsted Press, New York, 1976.

13.
H. Exton, New generating relations for products of two Laguerre polynomials, Indian J. Pure Appl. Math. 24 (1993), no. 6, 401-408.

14.
E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.

15.
L. J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, Cambridge, London, and New York, 1960.

16.
L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, London, and New York, 1966.

17.
H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.

18.
H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1985.

19.
H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1984.

20.
H. M. Srivastava and R. Panda, An integral representation for the product of two Jacobi polynomials, J. London Math. Soc. (2) 12 (1976), no. 4, 419-425.