  In this paper we resolve a question by Bringmann, Lovejoy, and Rolen on a new
vector-valued $U$-type function. We obtain an expression for a corresponding
family of Hecke-Appell-type sums in terms of mixed mock modular forms; that is,
we express the sum in terms of Appell functions and theta functions. This
$U$-type function appears from considering the special polynomials related to
generating functions for the partitions occurring in Gordon generalization of
the Rogers-Ramanujan identities.
