Truncated Hypergeometric Functions and Discretized Integrals

TL;DR

This paper introduces a finite truncation of hypergeometric series, provides a discretized integral representation, and establishes new formulas connecting truncated multiple polylogarithms with hypergeometric series, advancing the understanding of special functions.

Contribution

It presents a novel truncation method for hypergeometric functions and derives formulas linking truncated series with discretized integrals and multiple polylogarithms.

Findings

Established a discretized integral representation for truncated hypergeometric series

Proved Ohno-Zagier type formulas relating truncated multiple polylogarithms and hypergeometric series

Connected recent results on truncated series and discretized integrals from multiple zeta values

Abstract

We introduce a kind of finite truncation of the hypergeometric series and provide its discretized integral representation. This is motivated by recent results of Maesaka-Seki-Watanabe and Hirose-Matsusaka-Seki on the identity between truncated series and discretized integrals which comes from the mutliple zeta values and multiple polylogarithms. We also prove the formula of Ohno-Zagier type which relates the truncated multiple polylogarithms and the truncated hypergeometric series.