Truncated MacMahon-type $q$-series in arithmetic progressions and Chebyshev expansions

Abstract

We study three finite families of MacMahon-type $q$-series. The first is the arithmetic-progression family $B_{k,m,N,r}^{\pm }(q)$, which contains the usual truncated MacMahon series and their odd-part analogues as special cases. The other two families, $A_{k,m,a}(q)$ and $C_{k,m,a}(q)$, replace the squared linear factors by quadratic factors and are naturally controlled by Chebyshev polynomials. We prove a double $q$-binomial expansion for $B_{k,m,N,r}^{\pm }(q)$ by a weighted elementary-symmetric-function argument; this proves and extends Merca's Conjecture 7 on truncated MacMahon series. We also establish finite Chebyshev polynomial expansions for $A_{k,m,a}(q)$ and $C_{k,m,a}(q)$, and we give weighted partition proofs of the $a=-2$ identities for both the ordinary family $A_{k,m,a}(q)$ and the odd family $C_{k,m,a}(q)$.

19 pages