Continuous Hahn polynomials

In mathematics, the continuous Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by In mathematics, the continuous Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. Closely related polynomials include the dual Hahn polynomials Rn(x;γ,δ,N), the Hahn polynomials Qn(x;a,b,c), and the continuous dual Hahn polynomials Sn(x;a,b,c). These polynomials all have q-analogs with an extra parameter q, such as the q-Hahn polynomials Qn(x;α,β, N;q), and so on. The continuous Hahn polynomials pn(x;a,b,c,d) are orthogonal with respect to the weight function In particular, they satisfy the orthogonality relation for ℜ ( a ) > 0 {displaystyle Re (a)>0} , ℜ ( b ) > 0 {displaystyle Re (b)>0} , ℜ ( c ) > 0 {displaystyle Re (c)>0} , ℜ ( d ) > 0 {displaystyle Re (d)>0} , c = a ¯ {displaystyle c={overline {a}}} , d = b ¯ {displaystyle d={overline {b}}} . The sequence of continuous Hahn polynomials satisfies the recurrence relation The continuous Hahn polynomials are given by the Rodrigues-like formula The continuous Hahn polynomials have the following generating function: