Solving connection and linearization problems within the Askey scheme and its q-analogue via inversion formulas

Abstract

For the polynomial families { P n ( x)} n belonging to the Askey scheme or to its q-analogue, the hypergeometric representation provides a natural expansion of the form P n ( x)=∑ m=0 n D m ( n) θ m ( x), where the expanding basis θ m ( x) is, in general, a product of Pochhammer symbols or q-shifted factorials. In this paper we solve the corresponding inversion problem, i.e. we compute the coefficients I m ( n) in the expansion θ n ( x)=∑ m=0 n I m ( n) P m ( x), which are then used as a tool for solving any connection and linearization problem within the Askey scheme and its q-analogue. Extensions of this approach for polynomials outside these two schemes are also given.