Some results on generalized multiple fractional part integrals

Abstract

In this paper, the following multiple fractional part integrals $I_{n,m}^{p_1,p_2,\dots ,p_n}={\int }_{[0,1{]}^n}\prod _{j=1}^n{x}_j^{p_j}\{S_n^{-1}{\}}^m\mathrm{d}{x}_1\cdots \mathrm{d}{x}_n$ and $J_{n,m}^p={\int }_{[0,1{]}^n}{S}_n^p\{S_n^{-1}{\}}^m\mathrm{d}{x}_1\cdots \mathrm{d}{x}_n$ are calculated for non-negative integers $p_1,p_2,\dots ,p_n,p,m$ and positive integer n, where $\left\{u\right\}$ denotes the fractional part of u and $S_n=x_1+x_2+\cdots +x_n.$ It is proved that $I_{1,m}^k$ has a closed form for non-negative integer $k\ge m$. We also prove that $I_{n,m}^{p_1,p_2,\dots ,p_n}$ ($n=2$ and $n=3$) and $J_{n,m}^p$ can be expressed as linear combinations of the Riemann zeta function, logarithmic function and some binomial coefficients. In particular, $J_{n,m}^p$ has a closed form for $p+n-1\ge m$. Moreover, some identities and recursive formulas of the above integrals are obtained.

Keywords:

• multiple integrals
• fractional part
• Riemann zeta function

2010 Mathematics Subject Classifications:

• 26A42
• 26B15
• 40A30

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This paper was supported by National Natural Science Foundation of China [grant number 61379009].