Zeta-star Values Research Articles

SOME IDENTITIES IN THE HARMONIC ALGEBRA CONCERNED WITH MULTIPLE ZETA VALUES

Let 𝔄 be a commutative ℚ-algebra, A be an alphabet of non-commutative letters and 𝔥1 be the non-commutative polynomial algebra generated by the set A over the algebra 𝔄. The harmonic algebra 𝔥1 is closely related to multiple zeta values and multiple zeta-star values in some special case. Some identities in the algebra 𝔥1 are given.

ON HOFFMAN'S RELATION FOR MULTIPLE ZETA-STAR VALUES

The multiple zeta values are natural generalization of Riemann zeta values, which are real numbers first considered by Euler. It is known that there are many linear relations among these values with rational coefficients. Hoffman [Multiple harmonic series, Pacific J. Math.152 (1992) 275–290] gives a class of relations among multiple zeta values, which is called Hoffman's relation. In this paper, we show the corresponding relation for multiple zeta-star values by using partial-fractions expansion as Hoffman did.

On a conjecture of Kaneko and Ohno

Let $X_0^{\star}(k,n,s)$ denote the sum of all multiple zeta-star values of weight $k$, depth $n$ and height $s$. Kaneko and Ohno conjecture that for any positive integers $m,n,s$ with $m,n\geqslant s$, the difference $(-1)^mX_0^{\star}(m+n+1,n+1,s)-(-1)^nX_0^{\star}(m+n+1,m+1,s)$ can be expressed as a polynomial of zeta values with rational coefficients. We give a proof of this conjecture in this paper.

The Bowman–Bradley theorem for multiple zeta-star values

The Bowman-Bradley theorem asserts that the multiple zeta values at the sequences obtained by inserting a fixed number of twos between 3,1,...,3,1 add up to a rational multiple of a power of pi. We establish its counterpart for multiple zeta-star values by showing an identity in a non-commutative polynomial algebra introduced by Hoffman.

Multiple zeta values vs. multiple zeta-star values

We discuss an algebraic connection between two kinds of multiple zeta values or their q-analogues: the ( q-)multiple zeta values and the ( q-)multiple zeta-star values. These two classes of values generate the same algebra, but in this paper, we show that the translation map between these two classes has a quite interesting algebraic property in a general setting, for example, the compatibility with the harmonic product. We also study several applications of the result.

ON A KIND OF DUALITY OF MULTIPLE ZETA-STAR VALUES

A duality-type relation for height one multiple zeta-star values is established. A conjectural generalization to the case of arbitrary height is also presented.

On functions of Arakawa and Kaneko and multiple zeta values

We study two functions ?k(s) and ?k1?kr(s) introduced by Arakawa and Kaneko [3] and relate them with (finite) multiple zeta values and multiple zeta star values using elementary methods. In particular, we give an alternative proof of a result of Ohno [14].

On functions of Arakawa and Kaneko and multiple zeta values

We study two functions ξk(s) and ξk1…kr(s) introduced by Arakawa and Kaneko [3] and relate them with (finite) multiple zeta values and multiple zeta star values using elementary methods. In particular, we give an alternative proof of a result of Ohno [14].

An algebraic proof of the cyclic sum formula for multiple zeta values

We introduce an algebraic formulation of the cyclic sum formulas for multiple zeta values and for multiple zeta-star values. We also present an algebraic proof of cyclic sum formulas for multiple zeta values and for multiple zeta-star values by reducing them to Kawashima's relation.

A $q$-analogue of non-strict multiple zeta values and basic hypergeometric series

We consider the generating function for a q-analogue of non-strict multiple zeta values (or multiple zeta-star values) and prove an explicit formula for it in terms of a basic hypergeometric series 3 φ 2 . By specializing the variables in the generating function, we reproduce the sum formula obtained by Ohno and Okuda and get some relations in the case of full height.

Sum Relations for Multiple Zeta Values and Connection Formulas for the Gauss Hypergeometric Functions

We give an explicit representation for the sums of multiple zeta-star values of fixed weight and height in terms of Riemann zeta values.