Extensions of the Mehler-Weisner and other results for the Hermite function
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- by M. E. Cohen PDF
- Math. Comp. 30 (1976), 553-564 Request permission
Abstract:
The purpose of this paper is to present expansions which generalize some well-known formulae for the Hermite function. Among these are the Weisner [20] extension of Mehler’s [17] bilinear relation, some recent results of Carlitz [4], and the Bateman [2] addition theorem. A bilateral generating function involving the product of the Hermite and ultraspherical polynomials is given. Finally, some general polynomial expansion theorems are derived.References
- Richard Askey, Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR 0481145
- Harry Bateman, A partial differential equation connected with the functions of the parabolic cylinder, Bull. Amer. Math. Soc. 41 (1935), no. 12, 884–893. MR 1563210, DOI 10.1090/S0002-9904-1935-06214-7
- Fred Brafman, Some generating functions for Laguerre and Hermite polynomials, Canadian J. Math. 9 (1957), 180–187. MR 85363, DOI 10.4153/CJM-1957-020-1
- L. Carlitz, Some extensions of the Mehler formula, Collect. Math. 21 (1970), 117–130. MR 279354
- M. E. Cohen, On Jacobi functions and multiplication theorems for integrals of Bessel functions, J. Math. Anal. Appl. 57 (1977), no. 2, 469–475. MR 432945, DOI 10.1016/0022-247X(77)90273-6
- 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 442305, DOI 10.1137/0507053 A. ERDÉLYI, "Über eine Erzeugende Funktion von Produkten Hermitescher Polynome," Math. Z., v. 44, 1938, pp. 201-211.
- Ervin Feldheim, Développements en série de polynomes d’Hermite et de Laguerre à l’aide des transformations de Gauss et de Hankel. I et II, Nederl. Akad. Wetensch., Proc. 43 (1940), 224–248 (French). MR 1400
- H. W. Gould and A. T. Hopper, Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J. 29 (1962), 51–63. MR 132853 I. S. GRADŠTEĬN & L. M. RYŽTK, Table of Integrals. Series, and Products, 4th ed., Fizmatgiz, Moscow, 1963; English transl., Academic Press, New York and London, 1965. MR 28 #5198; 33 #5952.
- R. P. Gupta and G. C. Jain, A generalized Hermite distribution and its properties, SIAM J. Appl. Math. 27 (1974), 359–363. MR 350087, DOI 10.1137/0127027 G. H. HARDY, "Summation of a series of polynomials of Laguerre," J. London Math. Soc., v. 7, 1932, pp. 138-139. E. HILLE, "On Laguerre’s series. II," Proc Nat. Acad. Sci. U.S.A., v. 12, 1926, pp. 265-269. J. KAMPÉ de FÉRIET, Danske Vid. Selsk. Math. Fys. Medd., v. 5, 1923, no. 2.
- Eugene Lukacs, Characteristic functions, Griffin’s Statistical Monographs & Courses, No. 5, Hafner Publishing Co., New York, 1960. MR 0124075 Y. L. LUKE, The Special Functions and Their Approximations. Vols. 1,2, Math. in Sci. and Engineering, vol. 53, Academic Press, New York and London, 1969. MR 39 #3039; 40 #2909. F. G. MEHLER, "Ueber die Entwichlung einer Funktion von beliebig vielen Variablen nach Laplaceschen Funktionen höhrer Ordnung," J. Reine Angew. Math., v. 66, 1866, pp. 161-176.
- O. V. Sarmanov and Z. N. Bratoeva, Probabilistic properties of bilinear expansions in Hermite polynomials, Teor. Verojatnost. i Primenen. 12 (1967), 520–531 (Russian, with English summary). MR 0216541 G. N. WATSON, "Notes on generating functions of polynomials: (2) Hermite polynomials," J. London Math. Soc., v. 8, 1933, pp. 194-199.
- Louis Weisner, Generating functions for Hermite functions, Canadian J. Math. 11 (1959), 141–147. MR 109903, DOI 10.4153/CJM-1959-018-4
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Math. Comp. 30 (1976), 553-564
- MSC: Primary 33A65
- DOI: https://doi.org/10.1090/S0025-5718-1976-0419894-3
- MathSciNet review: 0419894