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A Nearly Forgotten Formula: Halphen’s Identity for Derivatives and Beyond

2021; Taylor & Francis; Volume: 94; Issue: 4 Linguagem: Inglês

10.1080/0025570x.2021.1951537

1930-0980

Ulrich Abel,

Analytic Number Theory Research

Summary–We bring to light a nearly forgotten identity found by Halphen in 1880. A special case is an expression for the nth derivative of exp (1/x). This formula was studied by Daboul, Mangaldan, Spivey, and Taylor in their recent paper in this Magazine. We reproduce Halphen’s original proof in a slightly improved form, which verifies his identity for monomial functions. More generally, we study derivatives of f(xα) and establish a formula for the higher order derivatives of the product f(xα)g(xβ). As a consequence, we obtain a new proof of Halphen’s result which establishes the identity for arbitrary functions of sufficient smoothness.