Some vanishing sums involving binomial coefficients in the denominator
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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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  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  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 ,  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|
|Page Range:||pp. 27-32|
|Access rights to Published version:||Open Access|
|References:|| W.C.Horrace - On the di�erence of maxima from independent uniform samples and a hy- pergeometric identity, Preprint.  M.Petkovsek, H.S.Wilf and D.Zeilberger - \A=B", A.K.Peters 1996.  B.Sury - Sum of the reciprocals of the binomial coe�cients, European J. Combin. 14 (1993) 351-353.  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|
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