IJPAM: Volume 105, No. 4 (2015)

ON THE INTERESTING IDENTITY $\sum_{j=0}^{2016}{2016\choose j}(-1)^j(2015^{2015}-j)^{2016}=2016!$

Pavel Trojovský
Department of Mathematics
Faculty of Science
University of Hradec Králové
Rokitanského 62
50003 Hradec Králové, CZECH REPUBLIC

Abstract. We shall apply combinatorial arguments to provide the following identity relating factorial, binomial numbers and polynomial values at complex points: let $P(x)\in \mathbb{C}[x]$ be a polynomial with degree $k\geq 0$ and leading coefficient $b_k$. Then

$\displaystyle\sum_{j=0}^{k}{k\choose j}(-1)^jP(z-j)=b_kk!$, for all $z\in \mathbb{C}$.
As an immediate consequence, we obtain the identity in the title.

Received: September 4, 2015

AMS Subject Classification: 11A25, 05A10

Key Words and Phrases: factorial, binomial, combinatorics

Download paper from here.

DOI: 10.12732/ijpam.v105i4.12 How to cite this paper?

International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2015
Volume: 105
Issue: 4
Pages: 723 - 726

$\sum_{j=0}^{2016}{2016\choose j}(-1)^j(2015^{2015}-j)^{2016}=2016!$%22&as_occt=any&as_epq=&as_oq=&as_eq=&as_publication=&as_ylo=&as_yhi=&as_sdtAAP=1&as_sdtp=1" title="Click to search Google Scholar for this entry" rel="nofollow">Google Scholar; DOI (International DOI Foundation); WorldCAT.

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).