Interpolation of \(q\)-analogue of multiple zeta and zeta-star values (Q503692)

Let $k_1,\ldots,k_d$ be positive integers with $k_1\geq 2$. The multiple zeta values \[ \zeta(k_1,\ldots,k_d)=\sum_{m_1>\ldots>m_d>0}\frac{1}{m_1^{k_1}\ldots m_d^{k_d}},\tag{1} \] as well as their variant, the multiple zeta star values \[ \zeta^{\star}(k_1,\ldots,k_d)=\sum_{m_1\geq\ldots\geq m_d>0}\frac{1}{m_1^{k_1}\ldots m_d^{k_d}},\tag{2} \] have $q$-analogues $\zeta_q(k_1,\ldots,k_n)$, $\zeta_q^{\star}(k_1,\ldots,k_d) \in \mathbb Q[[q]]$ which for $q=1$ specialize to (1), respectively (2). An important property is that they share at least some of the algebraic properties of multiple zeta (star) values, for example they are known to satisfy a variant of the Kawashima relations, [\textit{Y. Takeyama}, Ramanujan J. 27, No. 1, 15--28 (2012; Zbl 1305.05021)]. On the other hand, \textit{S. Yamamoto} [J. Algebra 385, 102--114 (2013; Zbl 1336.11062)], introduced interpolation polynomials $\zeta^t(k_1,\ldots,k_d) \in \mathbb \mathcal \mathbb R[t]$ which specialize to (1), (2) for $t=0$, $t=1$ respectively. Again, an analogue of the Kawashima relations is known to hold in this context, [\textit{T. Tanaka} and \textit{N. Wakabayashi}, J. Algebra 447, 424--431 (2016; Zbl 1370.11104)]. In the present paper under review, bringing the two strands together, the author introduces so-called $t$-$q$-multiple zeta values (or $t$-$q$MZVs for short) $\zeta^t_q(k_1,\ldots,k_d) \in \mathbb Q[[q]][t]$ which interpolate between $\zeta_q(k_1,\ldots,k_d)$ and $\zeta_q^{\star}(k_1,\ldots,k_d)$. The main result (Theorem 2) is that $t$-$q$MZVs satisfy Kawashima type relations which specialize to Takeyama's for $t=0$ and $t=1$, as well as to Tanaka-Wakabayashi's for $q=1$. Using Theorem 2, the author also proves a cyclic sum formula (Theorem 3) and a Hoffman type relation (Theorem 13) for $t$-$q$MZVs.