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 is an element of C are non-zero and defined such that vertical bar q vertical bar < 1 and vertical bar b/a vertical bar < vertical bar z vertical bar < 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 >= 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 bar right arrow 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 a is an element of 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. (C) 2017 Elsevier Inc. All rights reserved.