Interpolated multiple $t$-values of general level with fixed weight, depth and height
Zhonghua Li, Zhenlu Wang
TL;DR
This work introduces the interpolated multiple $t$-values of general level $N$, $t_{N,a}^r$, as a unifying interpolation between non-star and star MtVs. It derives a generating function for sums of these interpolated MtVs with fixed weight, depth, and height, and expresses this generating function via a generalized hypergeometric function $_{3}F_{2}$ evaluated at $1$, with parameters tied to formal variables $u,v,w$ and the interpolation parameter $r$. The main contribution is a comprehensive framework that recovers earlier results for MtVs, MtSVs, and interpolated MZVs in special cases, and extends them to height-restricted and weighted sums. This provides new closed forms, differential-equation methods, and explicit identities that deepen the connections among level-$N$ variants of zeta-like values and offer tools for further arithmetic and combinatorial investigations.
Abstract
In this paper, we introduce the interpolated multiple $t$-values of general level and represent a generating function for sums of interpolated multiple $t$-values of general level with fixed weight, depth, and height in terms of a generalized hypergeometric function $_3F_2$ evaluated at $1$. Furthermore, we explore several special cases of our results. The theorems presented in this paper extend earlier results on multiple zeta values and multiple $t$-values of general level.