Communications in Number Theory and Physics

Volume 13 (2019)

Number 3

An odd variant of multiple zeta values

Pages: 529 – 567



Michael E. Hoffman (Department of Mathematics, United States Naval Academy, Annapolis, Maryland, U.S.A.)


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.


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.

Received 30 August 2018

Accepted 25 January 2019

Published 8 August 2022