کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4637892 | 1631989 | 2016 | 14 صفحه PDF | دانلود رایگان |
In this paper, the following multiple fractional part integrals In,βα1,α2,⋯,αn=∫[0,1]n∏j=1nxjαj{Sn−1}βdx1⋯dxn and Jn,βα=∫[0,1]nSnα{Sn−1}βdx1⋯dxn are studied for positive integer nn and complex values of α,β,αj(j=1,2,⋯,n)α,β,αj(j=1,2,⋯,n), where {u}{u} denotes the fractional part of uu, R(s)R(s) denotes the real part of ss and Sn=x1+x2+⋯+xnSn=x1+x2+⋯+xn. It is proved that I1,βα can be represented as a linear combination of the Riemann zeta function, the Beta function and Euler’s constant as R(β)>−1R(β)>−1. Moreover, In,βα1,α2,⋯,αn can be expressed by In−1,βα1,α2,⋯,αn−1, the Beta function and the incomplete Beta function for n=2,3n=2,3. In addition, the recurrence formula of Jn,βα(n=2,3,⋯) is established and Jn,βα can be expressed by I1,βα, logarithmic function and some binomial coefficients.
Journal: Journal of Computational and Applied Mathematics - Volume 302, 15 August 2016, Pages 186–199