Some properties of special functions derived from the theory of continuous transformation groups
HTML articles powered by AMS MathViewer
- by Mrinal Kanti Das PDF
- Proc. Amer. Math. Soc. 35 (1972), 565-573 Request permission
Abstract:
The theory of continuous transformation groups is utilized in the study of some properties of special functions. On constructing the continuous transformation groups corresponding to a suitably defined infinitesimal transformation, a bilateral generating relation involving Laguerre polynomials $\{ L_n^{(\alpha )}(x)\}$ is obtained in $\S 2$. It is shown to be a generalisation of Brafman’s result. In the last section raising and lowering operators for $\{ P_n^{(\alpha ,\beta - n)}(x)\}$ and their commutator are introduced and on showing that they generate a 3-dimensional Lie algebra, the idea of c.t. groups is employed to establish a generating relation involving $\{ P_n^{(\alpha ,\beta - n)}(x)\}$ which is seen to yield a number of known results. Moreover, a bilateral generating relation involving $\{ P_n^{(\alpha ,\beta - n)}(x)\}$ is obtained; this is seen to be a generalisation of a well-known relation due to Weisner.References
- Bruria Kaufman, Special functions of mathematical physics from the viewpoint of Lie algebra, J. Mathematical Phys. 7 (1966), 447–457. MR 197788, DOI 10.1063/1.1704953
- Santi Kumar Chatterjea, Quelques fonctions génératrices des polynômes d’Hermite, du point de vue de l’algèbre de Lie, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A600–A602 (French). MR 240360 M. K. Das, Sur les polynômes de Laguerre, du point de vue de l’algèbre de Lie, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A380-A383. MR 41 #2093. —, Sur les polynômes d’Hermite, du point de vue de l’algèbre de Lie, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A452-A455. MR 41 #3845.
- Mrinal Kanti Das, Sur les polynômes de Bessel, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A408–A411 (French). MR 271417
- V. K. Varma, Double hypergeometric functions as generating functions of the Jacobi and Laguerre polynomials, J. Indian Math. Soc. (N.S.) 32 (1968), 1–5. MR 241725
- Earl D. Rainville, Special functions, The Macmillan Company, New York, 1960. MR 0107725
- Louis Weisner, Group-theoretic origin of certain generating functions, Pacific J. Math. 5 (1955), 1033–1039. MR 86905
- 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
- G. P. Srivastava, Some bilinear generating functions, Proc. Nat. Acad. Sci. U.S.A. 64 (1969), 462–465. MR 267150, DOI 10.1073/pnas.64.2.462
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 35 (1972), 565-573
- MSC: Primary 33A75
- DOI: https://doi.org/10.1090/S0002-9939-1972-0302979-8
- MathSciNet review: 0302979