Continued fractions and q-series generating functions for the generalized sum-of-divisors functions

We construct new continued fraction expansions of Jacobi-type J-fractions in z whose power series expansions generate the ratio of the q-Pochhammer symbols, (a;q)n/(b;q)n, for all integers n≥0 and where a,b,q∈C are non-zero and defined such that |q|<1 and |b/a|<|z|<1. If we set the parameters (a,b):=(q,q2) in these generalized series expansions, then we have a corresponding J-fraction enumerating the sequence of terms (1−q)/(1−qn+1) over all integers n≥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↦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, σα(n), when α∈Z+. To expand the new q-series generating functions for these special arithmetic functions we define a generalized class 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.