Some classes of generating functions for the Laguerre and Hermite polynomials

Author:
M. E. Cohen

Journal:
Math. Comp. **31** (1977), 511-518

MSC:
Primary 33A65

DOI:
https://doi.org/10.1090/S0025-5718-1977-0442323-1

MathSciNet review:
0442323

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In the first half of the article, we present two theorems which give, as special cases, a number of new classes of generating functions for the Laguerre polynomial. These formulae extend the recent results of Carlitz [2] and others. The latter part of our work deals with two theorems involving new generating functions for the Hermite and generalized Hermite polynomials, thus generalizing some well-known expansions. The method of proof adopted in this paper differs from that of previous workers.

**[1]**J. W. Brown,*On zero type sets of Laguerre polynomials*, Duke Math. J.**35**(1968), 821–823. MR**0234027****[2]**L. Carlitz,*Some generating functions for Laguerre polynomials*, Duke Math. J.**35**(1968), 825–827. MR**0240351****[3]**S. K. Chatterjea,*Some generating functions*, Duke Math. J.**32**(1965), 563–564. MR**0181762****[4]**M. E. Cohen,*On expansion problems: new classes of formulas for the classical functions*, SIAM J. Math. Anal.**7**(1976), no. 5, 702–712. MR**0442305**, https://doi.org/10.1137/0507053**[5]**M. E. Cohen,*Generating functions for the Jacobi polynomial*, Proc. Amer. Math. Soc.**57**(1976), no. 2, 271–275. MR**0404725**, https://doi.org/10.1090/S0002-9939-1976-0404725-X**[6]**Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,*Higher transcendental functions. Vols. I, II*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR**0058756****[7]**H. W. Gould and A. T. Hopper,*Operational formulas connected with two generalizations of Hermite polynomials*, Duke Math. J.**29**(1962), 51–63. MR**0132853****[8]**R. P. Gupta and G. C. Jain,*A generalized Hermite distribution and its properties*, SIAM J. Appl. Math.**27**(1974), 359–363. MR**0350087**, https://doi.org/10.1137/0127027**[9]**Yudell L. Luke,*Mathematical functions and their approximations*, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR**0501762****[10]**David Zeitlin,*A new class of generating functions for hypergeometric polynomials.*, Proc. Amer. Math. Soc.**25**(1970), 405–412. MR**0264123**, https://doi.org/10.1090/S0002-9939-1970-0264123-3

Retrieve articles in *Mathematics of Computation*
with MSC:
33A65

Retrieve articles in all journals with MSC: 33A65

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1977-0442323-1

Keywords:
Generating functions,
Hermite polynomial,
hypergeometric function,
incomplete gamma function,
Laguerre polynomial,
operators

Article copyright:
© Copyright 1977
American Mathematical Society