The extended Zeilberger algorithm with parameters

For a hypergeometric series ∑ k f(k, a, b, . . . , c) with parameters a, b, . . . , c, Paule has found a variation of Zeilberger’s algorithm to establish recurrence relations involving shifts on the parameters. We consider a more general problem concerning several similar hypergeometric terms f1(k, a, b, . . . , c), f2(k, a, b, . . . , c), . . ., fm(k, a, b, . . . , c). We present an algorithm to derive a linear relation among the sums ∑ k fi(k, a, b, . . . , c) (1 ≤ i ≤ m). Furthermore, when the summand fi contains the parameter x, we can require that the coefficients be x-free. Such relations with x-free coefficients can be used to determine whether a polynomial sequence satisfies the three term recurrence and structure relations for orthogonal polynomials. The q-analogue of this approach is called the extended q-Zeilberger’s algorithm, which can be employed to derive recurrence relations on the Askey-Wilson polynomials and the q-Racah polynomials.