The Pfaff/Cauchy derivative identities and Hurwitz type extensions |
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Authors: | Warren P Johnson |
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Institution: | (1) Department of Mathematics, Bates College, Lewiston, Maine, 04240;(2) Present address: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania, 17837 |
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Abstract: | 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.
Dedicated to Dick Askey on the occasion of his 70th birthday.
2000 Mathematics Subject Classification Primary—05A19; Secondary—33C45 |
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Keywords: | Derivative identities Abel-Rothe identities Hurwitz type identities Lagrange’ s series Laguerre and Jacobi polynomials |
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