LAUSR

Abstract 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 , 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 ↦ 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 ∈ 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.