Restricted divisor functions and half Appell sums in higher level

We examine the value distributions of coefficients in certain [Formula: see text]-series related to half Appell sums in higher level and the first moment of the Garvan’s [Formula: see text]-rank of partitions. We prove that these coefficients equal certain restricted divisor functions and can take any nonnegative integer value infinitely many times. As applications, we confirm a conjecture of Xiong on the coefficients of a half Lerch sum and a conjecture of Garvan and Jennings-Shaffer on the nonnegativity of spt-crank-type partitions.