Über eine Anwendung der L<scp>AGRANGES</scp>chen Reihe

Tinhofer, Gottfried

doi:10.1002/mana.19730580113

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Introduction: the Pfaff/cauchy Identities1

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1975

2007

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Article2

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“…Gould unearthed Cauchy's paper and Walker's first paper in 1972 [12], and Ruscheweyh [38] rediscovered (1.1) shortly thereafter. In the same issue of Mathematische Nachrichten as [38], Tinhofer [43] published an identity which, as Gould pointed out in Math Reviews, is the case v = u s of (1.4). In spite of Gould's efforts (he also reviewed [38]), and two more recent notes by Alzer [3,4], these identities still failed to attract much attention.…”

Section: Introduction: the Pfaff/cauchy Identitiesmentioning

confidence: 96%

The Pfaff/Cauchy derivative identities and Hurwitz type extensions

Johnson

2007

Ramanujan J

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More than 200 years ago, Pfaff found two generalizations of Leibniz's rule for the nth derivative of a product of two functions. Thirty years later Cauchy found two similar identities, one equivalent to one of Pfaff's and the other new. We give simple proofs of these little-known identities and some further history. We also give applications to Abel-Rothe type binomial identities, Lagrange's series, and Laguerre and Jacobi polynomials. Most importantly, we give extensions that are related to the Pfaff/Cauchy theorems as Hurwitz's generalized binomial theorems are to the Abel-Rothe identities. We apply these extensions to Laguerre and Jacobi polynomials as well.

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