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Identities involving reciprocals of binomial coefficients. (English) Zbl 1069.11008

In the previous paper [B. Sury, “Sum of the reciprocals of the binomial coefficients”, Eur. J. Comb. 14, No. 4, 351–353 (1993; Zbl 0783.05002)], the first author showed that \[ \sum_{r=0}^n {{1}\over {{n}\choose {r}}}={{n+1}\over {2^n}}\sum_{r=0}^n {{2^r}\over {n+1}}. \] This paper continues the above work and contains several very interesting identities giving short forms for certain finite and infinite sums of reciprocals of binomial coefficients of which two examples are \[ \sum_{r=m}^n {{(-1)^r}\over {{n\choose r}}}=\biggl((-1)^n+{{(-1)^m}\over {{n+1}\choose {m}}}\biggr){{n+1}\over {n+2}}, \] and \[ \sum_{r=m}^{\infty}{{1}\over {{n+r}\choose {r}}}={{n}\over {(n-1){{m+n-1} \choose{n-1}}}}. \] The method of generating such identities is to start with the observation that \[ {{1}\over {{{n}\choose {r}}}}=(n+1)\int_0^1 t^r(1-t)^{n-r} \,dt \] in order to reduce the computation of suitably chosen polynomials and power series whose coefficients involve reciprocals of binomial coefficients to the computation of the antiderivatives of the resulting functions, which, in turn, by specialization to suitable inputs give rise to the combinatorial identities.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
05A19 Combinatorial identities, bijective combinatorics

Citations:

Zbl 0783.05002
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