Recurrence equations and their classical continuous, discrete, and q-discrete orthogonal polynomial solutions

Classical orthogonal polynomials are solutions of a second order differential, difference, or q -difference equation for continuous, discrete, or q -discrete variables, respectively. Furthermore, all such systems satisfy a three-term recurrence equation of the form: aligned pₙ₊₁(x) = (Aₙ x + Bₙ)pₙ(x) - Cₙ pₙ₋₁(x), aligned p n + 1 ( x ) = ( A n x + B n ) p n ( x ) - C n p n - 1 ( x ) , for n≥ 0 n ≥ 0 with p₋₁=0,~p₀=1 p - 1 = 0 , p 0 = 1 . Given a holonomic three-term recurrence equation, we implement in and an algorithm which detects its classical orthogonal polynomial solutions for the continuous, discrete, and q -discrete variables when they exist. With our implementations, the results obtained using the Maple implementations by Koepf and Schmersau (Appl Math Comput 128:303–327, 2002) and Koorwinder and Swarttouw (Priv Commun, 1998) are easily recovered. In addition, we obtain new relations that extend beyond those previously established in the literature.