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Inversion techniques and combinatorial identities – strange evaluations of hypergeometric series. (English) Zbl 0815.05008

Let \(a_ i\) and \(b_ i\) denote two sequences of complex numbers such that the polynomial \(\varphi (x;n)= \prod_{k=1}^ n (a_ k+ xb_ k)\) is non-zero for natural numbers \(n\), \(x\) with the convention \(\varphi (x;0) =1\). H. W. Gould and L. C. Hsu [Duke Math. J. 40, 885- 891 (1973; Zbl 0281.05008)] obtained the following inverse pair: \[ \begin{aligned} f(n) &= \sum_{k=0}^ n (-1)^ k {n \choose k} \varphi(k; n)g(k) \qquad \text{ and}\\ g(n) &= \sum_{k=0}^ n (-1)^ k {n \choose k} {{a_{k+1}+ kb_{k+1}} \over {\varphi (n; k+1)}} f(k). \end{aligned} \] Inverse pairs are responsible in part of the proliferation of combinatorial identities: an identity somehow embedded in one element of the pair yields a sibling identity. The present paper reduces a number of strange evaluations of terminating hypergeometric series through the Gould-Hsu inverse pair to the Saalschütz formula. See also W. Chu [Forum Math. 7, No. 1, 117-129 (1995; Zbl 0815.05009)].

MSC:

05A19 Combinatorial identities, bijective combinatorics
33C20 Generalized hypergeometric series, \({}_pF_q\)
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