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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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Official URL: http://journals.aulonapress.com/index.php/ajm/arti...
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 | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Binomial coefficients | ||||
Journal or Publication Title: | Albanian Journal of Mathematics | ||||
Publisher: | Aulonna Press | ||||
ISSN: | 1930-1235 | ||||
Official Date: | 2008 | ||||
Dates: |
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Volume: | Vol.2 | ||||
Number: | No.1 | ||||
Page Range: | pp. 27-32 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Open Access (Creative Commons) | ||||
Date of first compliant deposit: | 1 August 2016 | ||||
Date of first compliant Open Access: | 1 August 2016 |
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