Generating Functions for the Ultraspherical Polynomials by Truesdell Method

Author(s): Srinivasulu Tadikonda and V. S. Bhagavan

Ultraspherical polynomials Cυ(x) n (Gegenbaur polynomials) are one of the most important families of orthogonal polynomials in mathematical chemistry, mathematical physics, probability theory, differential equations, combinatorics etc. One of the simplest ways to construct them is through their generating functions. In this paper, the generating functions for ultraspherical polynomials Cυ(x) n are obtained by using the Truesdell’s method giving a suitable interpretation to the index n. Further, a pair of linearly independent differential recurrence relations are used in order to derive generating functions for C υ (x). n The principal interest in our results lies in the fact that, how the Truesdell’s method is utilized in an effective and suitable way to Gegenbaur polynomials in order to derive two generating functions independently from ascending and descending recurrence relations, respectively. The generating functions, in turn yield, the Legendre polynomials as special case for . 2 υ = 1 The results are well known in the theory of special functions. Mathematics Subject Classification (2010): Primary 33C10, secondary 33C45, 33C80.

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