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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.)
##### Citations:
Zbl 0797.33013; Zbl 0823.33003
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