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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Some properties of special functions derived from the theory of continuous transformation groups


Author: Mrinal Kanti Das
Journal: Proc. Amer. Math. Soc. 35 (1972), 565-573
MSC: Primary 33A75
MathSciNet review: 0302979
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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 $ \S2$. 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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0302979-8
PII: S 0002-9939(1972)0302979-8
Article copyright: © Copyright 1972 American Mathematical Society