SOME VANISHING SUMS INVOLVING BINOMIAL COEFFICIENTS IN THE DENOMINATOR

S. Purkait; B. Sury

doi:10.51286/albjm/1204784959

2008 SOME VANISHING SUMS INVOLVING BINOMIAL COEFFICIENTS IN THE DENOMINATOR

S. Purkait, B. Sury

Author Affiliations +

Albanian J. Math. 2(1): 27-32 (2008). DOI: 10.51286/albjm/1204784959

Abstract

We obtain expressions for sums of the form ${\textstyle\sum_{j=0}^m}{(-1)}^j\frac{j^d\begin{pmatrix}m\\j\end{pmatrix}}{\begin{pmatrix}n+j\\j\end{pmatrix}}$ and deduce, for an even integer $d\geq0$ and $m=n>d/2$ , that this sum is $0$ or $\frac12$ according as to whether $d>0$ or not. Further, we prove for even $d>0$ that ${\textstyle\sum_{l=1}^d}c_{l-1}\frac{{(-1)}^l\begin{pmatrix}n\\l\end{pmatrix}l!}{(l+1)\begin{pmatrix}2n\\l+1\end{pmatrix}}=0$ where $c_r=\frac1{r!}{\textstyle\sum_{s=0}^r}{(-1)}^s\begin{pmatrix}r\\s\end{pmatrix}{(r-s+1)}^{d-1}$ . Similarly, we show when $d>0$ is even that ${\textstyle\sum_{r=0}^d}a_r\frac{r!\begin{pmatrix}n\\r+1\end{pmatrix}}{\begin{pmatrix}2n\\r+1\end{pmatrix}}=0$ , where $a_r=\frac{{(-1)}^{d+r}}{r!}{\textstyle\sum_{s=0}^r}{(-1)}^s\begin{pmatrix}r\\s\end{pmatrix}{(r-s+1)}^d$ .

Acknowledgements

We are indebted to William Horrace for communicating to us his identities which use probability theory and for pointing out (thanks to George Andrews) that they are special cases of the Chu-Vandermonde identities. We are also grateful to the referee who pointed out that some similar results due to A.Sofo appear in the paper titled ‘Sums of binomial coefficients in integral form’ published in the Proceedings of the 12th International Conference on Fibonacci numbers and their application in July 2006 - San Francisco, using different methods.

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S. Purkait. B. Sury. "SOME VANISHING SUMS INVOLVING BINOMIAL COEFFICIENTS IN THE DENOMINATOR." Albanian J. Math. 2 (1) 27 - 32, 2008. https://doi.org/10.51286/albjm/1204784959