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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. 2732. ISSN 19301235

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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 ChuVandermonde 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 wellknown [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:  19301235 
Date:  2008 
Volume:  Vol.2 
Number:  No.1 
Page Range:  pp. 2732 
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) 351353. [4] B.Sury, T.Wang and FZ.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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