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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

KENNETH S. WILLIAMS
KENNETH S. WILLIAMS
Affiliation:
Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6. e-mail: williams@math.carleton.ca
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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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