Partition identities and a theorem of Zagier
An infinite family of partition identities generalizing the well-known Eisenstien series identity \[ \sum_{\lambda=1 \atop \lambda \text{odd}}^{\infty} {(-1)^{(\lambda-1)/2} q^{\lambda} \over 1-q^{2\lambda}} =q \prod_{n=1}^{\infty} {(1-q^{8n})^4 \over (1-q^{4n})^2} \] is proven. The identities in this family are labelled by a positive integer \(m\) (\(m=1\) corresponding to the above identity) and express the infinite products \[ \prod_{n=1}^{\infty} {(1-q^{2(m+1)n})^{2m+2} \over (1-q^{(m+1)n})^{m+1}}, \qquad m\text{ odd}, \] and \[ {(q;q)_{\infty} \over (q^2;q^2)_{\infty}^2} {(1-q^{2(m+1)n})^{2m+2} \over (1-q^{(m+1)n})^{m+1}}, \qquad m\text{ even} \] in terms of the difference of two generating functions for partitions \(\Lambda\). Here \(\Lambda\) has \(\lfloor (m+1)/2\rfloor\) nonzero, distinct parts, satisfying some additional congruences. As a corollary of their \(q\)-series identities, the authors obtain expressions (in terms of certain partition functions) for \(T(n,k)\), the number of representations of \(n\) as a sum of \(k\) triangular numbers. The proof of the main result of the paper relies on a recent proof of \textit{D. Zagier} [Math. Res. Lett. 7, 597-604 (2000; Zbl 1125.11319)] of the Kac-Wakimoto conjecture for the affine denominator identity for the Lie superalgebra of type \(Q(m)\).
scientific article; zbMATH DE number 1888411
scientific article