A cubic transformation formula for the hypergeometric function 2F1(1/3, 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, ∑k=1 n-1 σ(k)σ(n - k) = 1/9σσ3(n), n ≡ 2(mod 3), k≡l(mod 3) is proved without the use of modular forms.

Additional Metadata
Persistent URL dx.doi.org/10.1017/S0305004104007832
Journal Mathematical Proceedings of the Cambridge Philosophical Society
Citation
Williams, K.S. (2004). A cubic transformation formula for2F1(1/3, 2/3;1;z) and some arithmetic convolution formulae. Mathematical Proceedings of the Cambridge Philosophical Society, 137(3), 519–539. doi:10.1017/S0305004104007832