Reduction formulas for higher order derivations and a hypergeometric identity

Abstract

A derivation (of order 1) satisfies the reduction formula $f(x^k) = kx^{k-1}f(x)$ for any integer k. In this article we find corresponding reduction formulas for derivations of higher order on commutative rings. More precisely, for every derivation f of order n and every positive integer k we find an explicit formula for $f(x^k)$ as a linear combination of $x^{k-1}f(x),x^{k-2}f(x^2), \ldots , x^{k-n}f(x^n)$. The proof hinges on the hypergeometric identity

$$\begin{aligned} \sum _{k \ge 0} (-1)^k \left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}n+d+1-k\\ n-j\end{array}}\right) \left( {\begin{array}{c}d+j-k\\ j\end{array}}\right) = \left( {\begin{array}{c}n\\ j\end{array}}\right) \end{aligned}$$

for any positive integer n, nonnegative integer d, and integer j satisfying $0 \le j \le n$. We prove this identity via the WZ-method.