The generalized Borwein conjecture. I: The Burge transform (Q2782420)

Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) defined by NEWLINE\[NEWLINE\prod^n_{k=1} (1- q^{3k- 2})(1- q^{3k- 1})= A_n(q^3)- qB_n(q^3)- q^2 C_n(q^3)NEWLINE\]NEWLINE have nonnegative coefficients. \textit{D. M. Bressoud} [Electron. J. Comb. 3, No. 2, Research paper R4 (1996; Zbl 0856.05007); printed version J. Comb. 3, 133-146 (1996)] observed that for NEWLINE\[NEWLINEG(N,M; \alpha,\beta, K)= \sum^{j=\infty}_{j=-\infty} (-1)^j q^{{1\over 2} Kj((\alpha+ \beta)j+ \alpha-\beta)}\Biggl[\begin{matrix} M+N\\ N- Kj\end{matrix}\Biggr],NEWLINE\]NEWLINE \(A_n(q)= G(n,n; 4/3,5/3,3)\), \(B_n(q)= G(n+1,n-1; 2/3,7/3,3)\), and \(C_n(q)= G(n+1,n- 1; 1/3,8/3, 3)\). Bressoud generalized Borwein's conjecture as follows: Let \(K\) be a positive integer and \(N\), \(M\), \(\alpha K\) and \(\beta K\) be nonnegative integers such that \(1\leq\alpha+ \beta\leq 2K-1\) (strict inequalities when \(K= 2\)) and \(\beta- K\leq N-M\leq K-\alpha\). Then \(G(N,M; \alpha,\beta, K)\) is a polynomial of \(q\) with nonnegative coefficients.NEWLINENEWLINENEWLINEThe paper uses the Burge transform to settle Bressoud's conjecture for two classes of problems. The first is when \(N= M\) with either \(\alpha\) or \(\beta\) being an integer, and the second is when \(N\neq M\) and both \(\alpha\) and \(\beta\) can be nonintegers.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00024].