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
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.
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This paper is dedicated to Ludwig Reich on the occasion of his 80th birthday.
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Ebanks, B., Kézdy, A.E. Reduction formulas for higher order derivations and a hypergeometric identity. Aequat. Math. 95, 1053–1065 (2021). https://doi.org/10.1007/s00010-021-00826-6
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DOI: https://doi.org/10.1007/s00010-021-00826-6
Keywords
- Derivation
- Derivation of higher order
- Commutative ring
- Binomial identity
- Hypergeometric identity
- Functional equation