The k-Rank Taylor Family: Canonical Harmonic Lifts, Principal Parts, and Rademacher Series

Abstract

For fixed $k\ge2$, Garvan's modified $k$-rank moments are encoded by odd elliptic Taylor coefficients of an odd-level Appell function. The completed coefficient of order $2n+1$ has weight $2n+3/2$, so the family cannot be realized as a fixed-weight vector-valued modular form. Nevertheless, all non-holomorphic parts arise from scalar contractions of canonical higher Serre derivatives of a single weight-$3/2$ vector-valued harmonic Maass form. Among lifts with the prescribed unary-theta shadow, the first $d=k-1$ Appell corrections are coordinates on the weakly holomorphic ambiguity and select a unique lift; the next correction is nonzero. The initial Taylor data recover the principal part without using positive Fourier coefficients of the modified moment series, and the principal part determines the shadow. The positive coefficients of the corresponding Maass--Poincaré lift have convergent Rademacher expansions; the canonical lift differs from it by a unique cusp form. This yields exact recursions for all even modified moments. For $k=2$, the cusp-form adjustment vanishes, giving an exact formula for the numbers of $2$-marked Durfee symbols involving a convergent Kloosterman--Bessel series.