Generating functions on extended Jacobi polynomials from Lie group view point
Let \(_2F_1(a,b;c,t)\) be the Gauss hypergeometries series. The extended Jacobi polynomials in this paper are defined by \(F_n(\alpha, \beta;x) =_2F_1 (-n,n+ \alpha+ \beta+1, \alpha+1; (x-a)/(b-a))\). The author introduces a Lie algebra of first order differential operators and calculates their exponential actions on the extended Jacobi polynomials. The author thus derives various generating functions of the polynomials. As some special and limiting cases the author recovers the generating functions of Laguerre, Hermite and Bessel polynomials. The differential operators in this paper are essentially the hypergeometric shift operators, which have been studied for Jacobi polynomials of several variables by \textit{G. J. Heckman} and \textit{E. M. Opdam} [An elementary approach to the hypergeometric shift operators of Opdam, Invent. Math. 103, 341-350 (1991; Zbl 0721.33009)].
scientific article; zbMATH DE number 911922