Formulae associated with 5, 7, 9 and 11 squares

Abstract

Let rk (n) denote the number of representations of n as the sum of k squares. We give elementary proofs of relations between rk (n) and rk (m) where n = 4λm and 4 ∤ m, when k = 5, 7, 9 and 11. The relations, which were first stated without proof by Stieltjes, are of the form rk (n) = Crk (m) where C depends on λ and on the residue of m modulo 8. They have recently been include by S. Cooper in a more complete description of the relations between rk (n) and rk (n′) where n′ is the squarefree part of n, when k = 5, 7, 9 and 11.