Further results on generalized multiple fractional part integrals for complex values

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.