An odd variant of multiple zeta values

For positive integers $i_1, \dotsc , i_k$ with $i_1 \gt 1$, we define the multiple $t$-value $t (i_1, \dotsc , i_k)$ as the sum of those terms of the usual infinite series for the multiple zeta value $\zeta (i_1, \dotsc, i_k)$ with odd denominators. Multiple $t$-values can be written as rational linear combinations of the alternating or “colored” multiple zeta values. Using known results for colored multiple zeta values, we obtain tables of multiple $t$-values through weight $7$, suggesting some interesting conjectures, including one that the dimension of the rational vector space generated by weight-$n$ multiple $t$-values has dimension equal to the $n\textrm{th}$ Fibonacci number. Like the multiple zeta values, the multiple $t$-values can be multiplied according to the rules of the harmonic algebra. Using this fact, we obtain explicit formulas for multiple $t$-values with repeated arguments analogous to those known for multiple zeta values. We express the generating function of the height one multiple $t$-values $t (n, 1, \dotsc, 1)$ in terms of a generalized hypergeometric function. We also define alternating multiple $t$-values and prove some results about them.

Keywords

multiple zeta values, multiple Hurwitz zeta function, colored multiple zeta values, quasi-symmetric functions, generalized hypergeometric function, Catalan’s constant, Dirichlet beta function

2010 Mathematics Subject Classification

Primary 11M32. Secondary 05E05, 11M35, 33C20.

Full Text (PDF format)

Received 30 August 2018

Accepted 25 January 2019

Published 8 August 2022