On functional equations of finite multiple polylogarithms

Abstract

Abstract. Recently, several people study finite multiple zeta values (FMZVs) and finite polylogarithms (FPs). In this paper, we introduce finite multiple polylogarithms (FMPs), which are natural generalizations of FMZVs and FPs, and we establish functional equations of FMPs. As applications of these functional equations, we calculate special values of FMPs containing generalizations of congruences obtained by Meštrović, Z. W. Sun, L. L. Zhao, Tauraso, and J. Zhao. We show supercongruences for certain generalized Bernoulli numbers and the Bernoulli numbers as an appendix. IntroductionFrom the end of twentieth century to the beginning of twenty-first century, Hoffman and J. Zhao had started research about mod p multiple harmonic sums, which are motivated by various generalizations of classical Wolstenholme's theorem. Recently, Kaneko and Zagier introduced a new "adélic" framework to describe the pioneer works by Hoffman and Zhao and they defined finite multiple zeta values (FMZVs). Let k 1 , . . . , k m be positive integers and k := (k 1 , . . . , k m ). Definition 1.1 (Kaneko and Zagier [10,11]). The finite multiple zeta value ζ A (k) is defined byand the finite multiple zeta-star value ζ ⋆ A (k) is defined byHere, the Q-algebra A is defined bywhere p runs over all prime numbers.In this framework, Kaneko and Zagier established a conjecture, which states that there is an isomorphism between the Q-algebra spanned by FMZVs and the quotient Q-algebra modulo the ideal generated by ζ(2) of the Q-algebra spanned by the usual multiple zeta values.On

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