An odd variant of multiple zeta values

TL;DR

This paper introduces multiple t-values, a variant of multiple zeta values focusing on odd denominators, explores their algebraic properties, explicit formulas, and conjectures about their dimensional structure, connecting them to colored multiple zeta values.

Contribution

It defines multiple t-values, derives explicit formulas, and conjectures their dimensionality relates to Fibonacci numbers, expanding the understanding of multiple zeta value variants.

Findings

Multiple t-values can be expressed as rational combinations of colored multiple zeta values.

Explicit formulas for repeated arguments of multiple t-values are derived.

The dimension of the space generated by weight-n multiple t-values may equal the nth Fibonacci number.

Abstract

For positive integers $i_1,...,i_k$ with $i_1 > 1$, we define the multiple $t$-value $t(i_1,...,i_k)$ as the sum of those terms in the usual infinite series for the multiple zeta value $\zeta(i_1,...,i_k)$ with odd denominators. 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 of repeated arguments analogous to those known for multiple zeta values. 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 $n$th Fibonacci number. We express the generating function of the height one multiple $t$-values $t(n,1,...,1)$ in terms of a generalized hypergeometric function. We also define alternating multiple $t$-values and prove some results about them.