Hypergeometric Orthogonal Polynomials

Abstract

In this chapter we deal with all families of hypergeometric orthogonal polynomials appearing in the Askey scheme on page 183. For each family of orthogonal polynomials we state the most important properties such as a representation as a hypergeometric function, orthogonality relation(s), the three-term recurrence relation, the second-order differential or difference equation, the forward shift (or degree lowering) and backward shift (or degree raising) operator, a Rodrigues-type formula and some generating functions. In each case we use the notation which seems to be most common in the literature. Moreover, in each case we mention the connection between various families by stating the appropriate limit relations. See also (Terwilliger in Lecture Notes in Mathematics, vol. 1883, Springer, Berlin, pp. 255–330, 2006) for an algebraic approach of this Askey scheme and (Temme and Lopez in Journal of Computational and Applied Mathematics 133:623–633, 2001) for a view from asymptotic analysis. For notations the reader is referred to chapter 1.