Identities for the Multiple Zeta (Star) Values

Abstract

In this paper we prove some new identities for multiple zeta values and multiple zeta star values of arbitrary depth by using the methods of integral computations of logarithm function and iterated integral representations of series. By applying the formulas obtained, we can prove that multiple zeta star values whose indices are the sequences (1, {1} m ,1) and (2, {1} m ,1) can be expressed polynomially in terms of zeta values, polylogarithms and ln(2). Finally, we also evaluate several restricted sum formulas involving multiple zeta values.

Repeated Integration and Explicit Formula for the $n$-th Integral of $x^m (\ln x)^{m'}$

“…The connection between repeated integrals and recurrent sum structures has been previously explored. In 2018, Ce Xu [10] illustrated the connection between similar integrals involving logarithms and a particular recurrent sum (the multiple zeta star function or multiple harmonic star sum). The multiple harmonic star sum (MHSS) [11,12,13] is defined as follows:…”

Explicit evaluation of some integrals involving polylogarithm functions

“…On the contrary, Freitas [1] gave explicit evaluations for J(−1, p, q) and K(r, 0, q) with r + q even. An anonymous reviewer told the author that Sofo [2,3,4]and Xu [5,6,7] had made many progresses in the area of integrals involving polylogarithm functions, which the author was initially unaware of. It is known to all that polylogarithmic functions are intrinsically connected with sums of harmonic numbers.…”

“…It is interesting that integrals of polylogarithm functions can be related to multiple zeta (star) values. By using integrals of polylogarithm functions, Xu [6] gave explicit expressions for some restricted multiple zeta (star) values. Some of lemmas used by Xu [6] were also re-discovered by the author in different forms.…”

“…By using integrals of polylogarithm functions, Xu [6] gave explicit expressions for some restricted multiple zeta (star) values. Some of lemmas used by Xu [6] were also re-discovered by the author in different forms. Furthermore, by using the iterated integral representation of multiple polylogarithm functions, Xu [7] proved some conjectures proposed by J. M. Borwein, D. M. Bradley and D. J. Broadhurst [9].…”

Evaluations of multiple polylogarithm functions, multiple zeta values and related zeta values

“…The motivation of this paper arises from the author's previous articles [19] and [20]. In [19,20], the author found many identities for alternating multiple zeta values and multiple zeta star values of arbitrary depth by using the methods iterated integral representations of series. multiple zeta values.…”