k秩泰勒族：典范调和提升、主部与拉德马赫级数
The k-Rank Taylor Family: Canonical Harmonic Lifts, Principal Parts, and Rademacher Series
AI总结：
该研究针对k≥2的Garvan修正k秩矩，构造k秩泰勒族的典范调和提升，推导其主部、拉德马赫展开及精确递归，k=2时得到2标记Durfee符号数的精确公式。
AI中文摘要：
对于固定的k≥2，Garvan修正k秩矩由奇级Appell函数的奇椭圆泰勒系数编码。阶数为2n+1的完成系数具有权2n+3/2，因此该族无法实现为固定权的向量值模形式。然而，所有非全纯部分均来自单个权为3/2的向量值调和Maass形式的典范高阶Serre导数的标量收缩。在具有指定单θ阴影的提升中，前d=k-1个Appell修正是弱全纯歧义的坐标，可选择唯一提升；下一个修正非零。初始泰勒数据无需使用修正矩级数的正傅里叶系数即可恢复主部，且主部决定阴影。对应Maass-Poincaré提升的正系数具有收敛的Rademacher展开；典范提升与它的差异为唯一尖点形式，这为所有偶修正矩产生精确递归。当k=2时，尖点形式调整消失，得到涉及收敛Kloosterman-Bessel级数的2标记Durfee符号数的精确公式。
英文摘要：
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.
