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[1][2][3] for
The sequence of continuous Hahn polynomials satisfies the recurrence relation[4] The continuous Hahn polynomials are given by the Rodrigues-like formula[5] The continuous Hahn polynomials have the following generating function:[6] A second, distinct generating function is given by