The study of relations between multiple zeta values (MZVs) has a long history, beginning perhaps with the famous correspondence between Goldbach and Euler. In recent years this area of research has been heavily influenced by the fast development in both quantum physics and computer science. Its importance is also reflected by the fact that the MZV relations (suitably regularized) are closely related to many different branches of mathematics.
In this paper, the author studies the interpolated multiple zeta values which can be regarded as interpolation polynomials of multiple zeta values and multiple zeta-star values.
For an index \(\mathbf{k}=(k_1,\dotsc,k_n)\) we define its depth \(\mathrm{dep}(\mathbf{k})=n\). For admissible \(\mathbf{k}\) (i.e., \(k_1\ge 2\)) and for a variable \(t\), the interpolated multiple zeta value (\(t\)-MZV) \(\zeta^t(\mathbf{k})\) is defined as \[ \zeta^t(\mathbf{k}):=\sum_{\substack{\mathbf{p}=(k_1 \Box k_2\Box \cdots\Box k_n) \\ \Box=\text{``,'' or ``\(+\)''} }} t^{n-\mathrm{dep}(\mathbf{p})} \zeta(\mathbf{p}). \] Note that \(\zeta^0(\mathbf{k})=\zeta(\mathbf{k})\) is the classical multiple zeta values and \(\zeta^1(\mathbf{k})=\zeta^\star(\mathbf{k})\) is the multiple zeta star values. Many relations, which are satisfied by MZVs and MZSVs simultaneously, are generalized to \(t\)-MZVs.
In this paper, the author proves the following main results: the shuffle regularized sum formula, a weighted sum formula (generalizing the famous sum weighted sum formula of L. Guo and B. Xie [J. Number Theory 129, No. 11, 2747–2765 (2009; Zbl 1229.11117)]). He also provides some evaluation formulas for interpolated multiple zeta values with even arguments. For example, he shows that for any positive integer \(k\) and non-negative integer \(n\) \[ \zeta^t(\{2k\}^n)=\sum_{\substack{i+j=n\\ i,j\ge0}} \zeta(\{2k\}^i) \zeta^\star(\{2k\}^j)(1-t)^i t^j. \] All the algebraic relations provided in this paper are deduced from the extended double shuffle relations.