Elliott's identity and hypergeometric functions |
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Authors: | R BalasubramanianS Ponnusamy M Vuorinen |
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Institution: | a The Institute of Mathematical Sciences, CIT Campus, Taramani, Madras 600 106, India b Department of Mathematics, Indian Institute of Technology, IIT-Madras, Chennai 600 036, India c Department of Mathematics, University of Helsinki, Yliopistonkatu 5, 00014 Helsinki, Finland |
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Abstract: | Elliott's identity involving the Gaussian hypergeometric series contains, as a special case, the classical Legendre identity for complete elliptic integrals. The aim of this paper is to derive a differentiation formula for an expression involving the Gaussian hypergeometric series, which, for appropriate values of the parameters, implies Elliott's identity and which also leads to concavity/convexity properties of certain related functions. We also show that Elliott's identity is equivalent to a formula of Ramanujan on the differentiation of quotients of hypergeometric functions. Applying these results we obtain a number of identities associated with the Legendre functions of the first and the second kinds, respectively. |
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Keywords: | Legendre's relation Elliott's identity and hypergeometric functions |
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