A Nearly Forgotten Formula: Halphen's Identity for Derivatives and Beyond. Issue 4 (8th August 2021)

• Record Type: Journal Article
• Authors: Abel, Ulrich
• Is Part Of: Mathematics magazine. Volume 94:Issue 4(2021)
• Page Start: 273
• Page End: 280
• Publication Date: 2021-08-08
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1080/0025570X.2021.1951537
• Languages: English
• ISSNs: 0025-570X

Abstract

Abstract : Summary– We bring to light a nearly forgotten identity found by Halphen in 1880. A special case is an expression for the n th 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 ( x α ) and establish a formula for the higher order derivatives of the product f ( x α ) g ( x β ) . As a consequence, we obtain a new proof of Halphen's result which establishes the identity for arbitrary functions of sufficient smoothness.