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A cubic transformation formula for $_{2}F_{1}(\frac{1}{3},\frac{2}{3};1;z)$ and some arithmetic convolution formulae
Published online by Cambridge University Press: 02 November 2004
Abstract
A cubic transformation formula for the hypergeometric function $_{2}F_{1}(\frac{1}{3},\frac{2}{3};1;z)$ is proved. As an application of this formula a number of arithmetic convolution sums are evaluated. For example, Melfi's formula, \[ \ukksum{n{-}1}{k{=}1}{k{\equiv}1 \, \mbox{(mod $3$)}}\kern-10pt\sigma(k)\sigma(n-k)=\frac{1}{9}\sigma_3(n),\;\;\;n \equiv 2\, \mbox{(mod $3$)}, \] is proved without the use of modular forms.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 137 , Issue 3 , November 2004 , pp. 519 - 539
- Copyright
- © 2004 Cambridge Philosophical Society
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