Title: A proof of the Kac-Wakimoto affine denominator formula for the strange series
Abstract: In [1], Kac and Wakimoto related various combinatorial formulas to the theory of root systems and representations of affine superalgebras.In particular, by specializing a conjectural affine denominator formula to the affine superalgebras associated to certain simple Lie superalgebras whose even parts are the simple Lie algebras of type A m , they obtain a beautiful conjectural formula (Conjecture 7.2 of [1]) which contains as special cases many classical identities from the theory of modular forms and elliptic functions, as well as many new identities.The purpose of this paper is to prove this conjecture.A statement in elementary terms (with no reference to affine algebras or root systems) is given as Theorem 3 in §3.The general case of the conjecture depends on m + 1 parameters q (which can be taken to be either a formal power series variable or a complex number of absolute value less than 1) and x 1 , . . ., x m (which are Laurent variables or nonzero complex numbers).Two specializations which are of particular interest, and which are singled out and stated as separate conjectures in [1], are the ones obtained by letting all variables x j tend to 1 or by taking m = 2.These two cases will be stated and proved separately in §1 and §2, respectively, since they have simpler and more appealing statements than the general case and since the ideas used to prove them give the essence of the general proof.The first of these two results, Theorem 1 below, gives a formula for the number of representations of an arbitrary non-negative integer as a sum of k triangular numbers when k has the form 4s 2 (corresponding to m = 2s -1) or 4s 2 + 4s (corresponding to m = 2s) for some positive integer s, the cases s = 1 being classical identities of Legendre for the number of representations of an integer as a sum of 4 or 8 triangular numbers.This theorem has also been proved by Milne [2].The second, Theorem 2 below, gives an infinite product expansion for the sum j,k>0, k odd q jk-1 (xy)The general identity, involving q, x 1 , . . ., x m , belongs naturally to the theory of Jacobi forms, i.e., if we write q = e 2πiτ and x j = e 2πiz j with τ ∈ H (upper half-plane) and z j ∈ C, then the functions appearing on its left-and righthand sides have modular transformation properties with respect to τ and elliptic transformation properties with respect to each z j .The modular properties with