Abstract:  For similar hypergeometric terms f₁(k), f₂(k),..., f_m(k), we present an algorithm to derive a linear relation among the sums ∑_kf_i(k) (1 ≤ i ≤ m). When the summand f_i(k) contains a parameter x, we further impose the condition that the coefficients of the linear relation are x-free. Such relations with x-free coefficients can be used to determine the structure relations for orthogonal polynomials and to derive recurrence relations for the connection coefficients between two classes of orthogonal polynomials. The extended Zeilberger algorithm can be easily adapted to basic hypergeometric terms. As examples, we use the algorithm or its q-analogue to establish linear relations among orthogonal polynomials and to derive recurrence relations with multiple parameters for hypergeometric sums and basic hypergeometric sums.

  AMS Classification:  33F10, 33C45, 33D45

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

  Download:   PDF