Abstract:  For a hypergeometric series ∑_kf(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),..., f_m(k,a,b,...,c). We present an algorithm to derive a linear relation among the sums ∑_kf_i(k,a,b,...,c) (1 ≤ i ≤ m). Furthermore, when the summand f_i 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.

  AMS Classification:  33F10, 33C45, 33D45

  Keywords:  Zeilberger's algorithm, the Gosper algorithm, hypergeometric series, orthogonal polynomials.

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