## Bhargava's Theorem

Let the th composition of a function be denoted , such that and . Denote , and define

 (1)

Let
 (2) (3) (4)

and
 (5) (6) (7)

Then if ,
 (8)

where and composition is done in terms of components.

Bhargava, S. On a Family of Ramanujan's Formulas for Sums of Fourth Powers.'' Ganita 43, 63-67, 1992.