# Who discovered the expansion for factorial as a successive difference of integers?

Who discovered the following theorem

$$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(n-r)^n=n!$$

• I'm willing to bet it was 'discovered' more than once. – Carl Witthoft Jul 31 '18 at 11:53

## 1 Answer

I firstly encountered this identity while working with related sums. Within this question on MSE, asked by myself, the user Robert Z refered to the particular case you are asking for as Tepper's Identity. In order to be precise Tepper's Identity is a more general result given by

$$n!~=~\sum_{k=0}^n(-1)^k\binom nk (a-k)^n,a\in\mathbb R$$

The aforementioned proof of Tepper's Identity was given by Ravi Prakash and was posted on the $$2$$nd of January $$2018$$ on the official website of the Romanian Mathematical Magazine which can be found here. Later on it reappeared within RMM - Famous Inequalities Marathon 1 - 100 as problem $$97$$. There are various proofs avaible nowadays, for instance Another Proof of Tepper's Identity by F. J. Papp published in the Mathematics Magazine Vol. $$45$$, No. $$3$$ $$($$May, $$1972)$$, pp. $$119-121$$ and A Proof of Tepper’s identity by Nguyen Trung Tuan posted on the $$10$$th of October $$2007$$.

Concerning the origin it seems like all sources are dating back to an arcticle called A Factorial Conjecture by Myron Tepper published in the Mathematics Magazine Vol. $$38$$, No. $$5$$ $$($$Nov., $$1965)$$, pp. $$303-304$$ $$($$the index to this volume can be found here$$)$$ proposing the identity, perhaps, for the first time.