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Bounds for the length of recurrence relations for convolutions of P-recursive sequences. (English) Zbl 0880.11017

Let \(S_r(n)\) denote the sum of the \(r\)th powers of the binomial coefficients \({n \choose k}\), where \(0\leq k\leq n\). Franel’s conjecture states that \(S_r(n)\) satisfies a linear recurrence of length \([{1\over 2} (r+1)]\), where the coefficients in the linear recurrence are polynomial functions of \(n\). The author proves Franel’s conjecture, as well as a more general theorem.

MSC:

11B37 Recurrences
11B65 Binomial coefficients; factorials; \(q\)-identities
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