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Some vanishing sums involving binomial coefficients in the denominator
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Purkait, S. (Soma) and Sury, B.. (2008) Some vanishing sums involving binomial coefficients in the denominator. Albanian Journal of Mathematics , Vol.2 (No.1). pp. 27-32. ISSN 1930-1235
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Abstract
Identities involving binomial coeffcients usually arise in situations where counting is carried out in two different ways. For instance, some identities obtained by William Horrace [1] using probability theory turn out to be special cases of the Chu-Vandermonde identities. Here, we obtain some generalizations of the identities observed by Horrace and give different types of proofs; these, in turn, give rise to some other new identities. In particular, we evaluate sums of the form Pm j=0 (1) j j d (mj) (n+jj ) and deduce that they vanish when d is even and m = n > d=2. It is well-known [2] that sums involving binomial coeffcients can usually be expressed in terms of the hypergeometric functions but it is more interesting if such a function can be evaluated explicitly at a given argument. Identities such as the ones we prove could perhaps be of some interest due to the explicit evaluation possible. The papers [3], [4] are among many which deal with identities for sums where the binomial coeffcients occur in the denominator and we use similar methods here.
| Item Type: | Journal Article |
|---|---|
| Subjects: | Q Science > QA Mathematics |
| Divisions: | Faculty of Science > Mathematics |
| Library of Congress Subject Headings (LCSH): | Binomial coefficients |
| Journal or Publication Title: | Albanian Journal of Mathematics |
| Publisher: | Aulonna Press |
| ISSN: | 1930-1235 |
| Date: | 2008 |
| Volume: | Vol.2 |
| Number: | No.1 |
| Page Range: | pp. 27-32 |
| Status: | Peer Reviewed |
| Publication Status: | Published |
| Access rights to Published version: | Open Access |
| References: | [1] W.C.Horrace - On the di�erence of maxima from independent uniform samples and a hy- pergeometric identity, Preprint. [2] M.Petkovsek, H.S.Wilf and D.Zeilberger - \A=B", A.K.Peters 1996. [3] B.Sury - Sum of the reciprocals of the binomial coe�cients, European J. Combin. 14 (1993) 351-353. [4] B.Sury, T.Wang and F-Z.Zhao - Identities involving reciprocals of binomial coe�cients, Jour- nal of Integer Sequences, Vol.7 (2004), Article 04.2.8 |
| URI: | http://wrap.warwick.ac.uk/id/eprint/49191 |
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