An extended scheme of classical special functions

Abstract

A unifying scheme of classical special functions of hypergeometric type obeying orthogonality or biorthogonality relations is described. It expands the Askey scheme of classical orthogonal polynomials and its $q$-analogue based on the Askey--Wilson polynomials. On the top, it has two-index biorthogonal functions formed from elliptic hypergeometric series with the absolutely continuous measure determined by the elliptic beta integral. A new result is an inclusion of complex hypergeometric functions into the scheme. Its further potential generalizations are discussed as well.

16 pp., written for a book dedicated to R. Askey and his influence on the theory of special functions