Continued Fractions and $q$-Series Generating Functions for the Generalized Sum-of-Divisors Functions

Abstract

We construct new continued fraction expansions of Jacobi-type J-fractions in $z$ whose power series expansions generate the ratio of the $q$-Pochhamer symbols, $(a; q)_n / (b; q)_n$, for all integers $n \geq 0$ and where $a,b,q \in \mathbb{C}$ are non-zero and defined such that $|q| < 1$ and $|b/a| < |z| < 1$. If we set the parameters $(a, b) := (q, q^2)$ in these generalized series expansions, then we have a corresponding J-fraction enumerating the sequence of terms $(1-q) / (1-q^{n+1})$ over all integers $n \geq 0$. Thus we are able to define new $q$-series expansions which correspond to the Lambert series generating the divisor function, $d(n)$, when we set $z \mapsto q$ in our new J-fraction expansions. By repeated differentiation with respect to $z$, we also use these generating functions to formulate new $q$-series expansions of the generating functions for the sums-of-divisors functions, $σ_α(n)$, when $α\in \mathbb{Z}^{+}$. To expand the new $q$-series generating functions for these special arithmetic functions we define a generalized classes of so-termed Stirling-number-like "$q$-coefficients", or Stirling $q$-coefficients, whose properties, relations to elementary symmetric polynomials, and relations to the convergents to our infinite J-fractions are also explored within the results proved in the article.