Let k and n be positive integers. Let sk(n) denote the number of representations ofn as the sum of k squares. Ramanujan [17], [18, p. 159] gave without proof a formulafor sk(n) when k is even. Mordell [15] used modular forms to give the first proof ofRamanujan's formula. In 2001 Cooper [6] used Ramanujan's 1ψ1 summation formulaand Jacobian elliptic functions and their derivatives to give a proof. It is our purpose toshow that when k is a multiple of 4, Ramanujan's formula can be proved in an entirelyelementary 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.

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Keywords Eisenstein series, Ramanujan's discriminantfunction, Sums of squares
Citation
Alaca, A, Alaca, S, & Williams, K.S. (2011). Sums of 4k squares: A polynomial approach. In Frontiers of Combinatorics and Number Theory (pp. 137–156).