Authors: Aycse Alaca, Saban Alaca and Kenneth S. Williams

Abstract: Let k and n be positive integers. Let s_{k}(n)denote the number of representations of n as the sum of k squares. Ramanujan [17], [18, p. 159] gave without proof a formula for s_{k}(n) when k is even. Mordell [15] used modular forms to give the first proof of Ramanujan’s formula. In 2001 Cooper [6] used Ramanujan’s 1ψ1 summation formula and Jacobian elliptic functions and their derivatives to give a proof. It is our purpose to show that when k is a multiple of 4, Ramanujan’s formula can be proved in an entirely elementary way using the properties of a certain class of polynomials. The values of sk(n) are determined explicitly for k = 4,8,12,16,20,24,28,32,36,40,44 and 48.


