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The Extended Zeilberger's Algorithm with Parameters | | William Y.C. Chen
; Qing-Hu Hou
; Yan-Ping Mu
; | | Date: |
10 Aug 2009 | | Abstract: | For a hypergeometric series $sum_k f(k,a, b, ...,c)$ 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 $f_1(k, a,
b,..., c)$, $f_2(k, a,b, ..., c)$, $...$, $f_m(k, a, b, ..., c)$. We present an
algorithm to derive a linear relation among the sums $sum_k f_i(k,a,b,...,c)$
$(1leq i leq m)$. 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. | | Source: | arXiv, 0908.1328 | | Services: | Forum | Review | PDF | Favorites | |
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