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 $n\ge 0$ 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 $n\ge 0$. 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, ${\sigma }_{\alpha }(n)$, when $\alpha \in {\mathbb{Z}}^{+}$. 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.

Keywords: divisor function; sum of divisors function; continued fraction; J-fraction. MSC Subject Class (2010): 11J70; 11Y65; 40A30; 11B65; 11A25