2009/08/10 by Chen, William Y. C., Hou, Qing-Hu, Mu, Yan-Ping · 1 citation
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.0908.1328
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.