Further results on generalized multiple fractional part integrals for complex values

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• 作者：Zhongfeng Sun ^{zfsun@sdut.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Aijuan Li ^{juanzi612@163.com" class="auth_mail" title="E-mail the corresponding author} ; Huizeng Qin ; ^{qin_hz@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Multiple integrals ; Fractional part ; Riemann zeta function ; Beta function ; Incomplete Beta function
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2016
• 出版时间：15 August 2016
• 年：2016
• 卷：302
• 期：Complete
• 页码：186-199
• 全文大小：441 K

文摘

In this paper, the following multiple fractional part integrals and are studied for positive integer n$n$ and complex values of α,β,α_j(j=1,2,⋯,n)$\alpha ,\beta ,{\alpha }_j(j=1,2,\cdots ,n)$, where {u}$\left\{u\right\}$ denotes the fractional part of u$u$, R(s)$\mathfrak{R}\left(s\right)$ denotes the real part of s$s$ and S_n=x₁+x₂+⋯+x_n$S_n=x_1+x_2+\cdots +x_n$. It is proved that i10" class="mathmlsrc">i10.gif&_user=111111111&_pii=S0377042716300619&_rdoc=1&_issn=03770427&md5=386318a410b6d771160e301f10a8511c">i10.gif"> can be represented as a linear combination of the Riemann zeta function, the Beta function and Euler’s constant as i11" class="mathmlsrc">i11.gif&_user=111111111&_pii=S0377042716300619&_rdoc=1&_issn=03770427&md5=63549c6fdb731afb217e4b231d8aaee5" title="Click to view the MathML source">R(β)>−1. Moreover, i12" class="mathmlsrc">i12.gif&_user=111111111&_pii=S0377042716300619&_rdoc=1&_issn=03770427&md5=4dcb4f0eaf77b075826b7142bd2c683a">i12.gif"> can be expressed by i13" class="mathmlsrc">i13.gif&_user=111111111&_pii=S0377042716300619&_rdoc=1&_issn=03770427&md5=a600e6354992ebbfb7a417655500588c">i13.gif">, the Beta function and the incomplete Beta function for i14" class="mathmlsrc">i14.gif&_user=111111111&_pii=S0377042716300619&_rdoc=1&_issn=03770427&md5=d2f4e97b4e800bef3840e3875e8883ef" title="Click to view the MathML source">n=2,3. In addition, the recurrence formula of i15" class="mathmlsrc">i15.gif&_user=111111111&_pii=S0377042716300619&_rdoc=1&_issn=03770427&md5=ba4c7698cb11941b046b7486433b7261">i15.gif"> is established and i16" class="mathmlsrc">i16.gif&_user=111111111&_pii=S0377042716300619&_rdoc=1&_issn=03770427&md5=fbfb807f5d6160c12d13a2a18afe11ed">i16.gif"> can be expressed by i10" class="mathmlsrc">i10.gif&_user=111111111&_pii=S0377042716300619&_rdoc=1&_issn=03770427&md5=386318a410b6d771160e301f10a8511c">i10.gif">, logarithmic function and some binomial coefficients.