## ASKEY SCHEME OF HYPERGEOMETRIC ORTHOGONAL POLYNOMIALS # Chapter 9 Hypergeometric Orthogonal Polynomials This chapter discusses all families of hypergeometric orthogonal polynomials in the Askey scheme on page 183. For each family, key properties are presented, including hypergeometric representation, orthogonality relations, three-term recurrence relations, differential or difference equations, shift operators, Rodrigues-type formulas, and generating functions. The notation used is standard in the literature. The connections between different families are explained through limit relations. References [500] and [498] provide an algebraic and asymptotic analysis approach, respectively. Notations are defined in chapter 1. ### 9.1 Wilson Hypergeometric Representation $$ \begin{aligned}&\frac{W_{n}(x^{2};a,b,c,d)}{(a+b)_{n}(a+c)_{n}(a+d)_{n}}\\ &=_{4}F_{3}\left(\begin{matrix}-n,n+a+b+c+d-1,a+ix,a-ix\\ a+b,a+c,a+d\end{matrix};1\right).\\ \end{aligned} $$## ASKEY SCHEME OF HYPERGEOMETRIC ORTHOGONAL POLYNOMIALS # Chapter 9 Hypergeometric Orthogonal Polynomials This chapter discusses all families of hypergeometric orthogonal polynomials in the Askey scheme on page 183. For each family, key properties are presented, including hypergeometric representation, orthogonality relations, three-term recurrence relations, differential or difference equations, shift operators, Rodrigues-type formulas, and generating functions. The notation used is standard in the literature. The connections between different families are explained through limit relations. References [500] and [498] provide an algebraic and asymptotic analysis approach, respectively. Notations are defined in chapter 1. ### 9.1 Wilson Hypergeometric Representation $$ \begin{aligned}&\frac{W_{n}(x^{2};a,b,c,d)}{(a+b)_{n}(a+c)_{n}(a+d)_{n}}\\ &=_{4}F_{3}\left(\begin{matrix}-n,n+a+b+c+d-1,a+ix,a-ix\\ a+b,a+c,a+d\end{matrix};1\right).\\ \end{aligned} $$