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On the inverse Legendre transform of a certain family of sequences. (English. Russian original) Zbl 1216.11027
Math. Notes 76, No. 2, 276-279 (2004); translation from Mat. Zametki 76, No. 2, 300-303 (2004).
Here the author proves the following theorem: For an integer \(r\geq2,\) let the sequence of numbers \(\{c_ {k}^ {(r)}(n)\}_ {k=0}^ \infty\), depending on the parameter \(n\), be given by the equation \[ \sum_ {k=0}^ n\left({n\atop k}\right)^ r\left({n+k\atop k}\right)^ r= \sum_ {k=0}^ n\left({n\atop k}\right)\left({n+k\atop k}\right)c_ k^ {(r)}(n),\;n=0,1,\dots \,. \] Then all the numbers \(c_ k^ {(r)}(n)\) are integers; they are calculated explicitly.
It is pointed out that the sequence \(\{c_ k^ {(r)}\}\) in the case \(r=2\) gives the inverse Legendre transform of the famous sequence of Apéry’s numbers expressing the denominators of the convergents in his proof of the irrationality of the number \(\zeta(3)=\sum_{n=1}^\infty n^{-3}\).
The problem, originally posed by A. L. Schmidt [J. Comput. Appl. Math. 49, No.1-3, 243–249 (1993; Zbl 0797.33013)] was solved by Schmidt for \(r=2\) and by V. Strehl [Discrete Math. 136, No. 1-3, 309–346 (1994; Zbl 0823.33003)] for \(r=2,3\).
MSC:
11B65 Binomial coefficients; factorials; \(q\)-identities
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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