Representation of a class of multiple fractional part integrals and their closed form

ABSTRACTIn this paper, the following generalized multiple fractional part integrals: In,μα1,α2,…,αn(a)=∫01∫01⋯∫01x1α1x2α2⋯xnαn1ax1x2⋯xnμ×dx1dx2⋯dxn, and k1,k2,…,knIn,μα1,α2,…,αn(a)=∫01∫01⋯∫011ax1x2⋯xnμ×∏i=1nxiαilnki⁡1xidx1dx2⋯dxn, are considered for complex numbers μ,αi (i=1,2,3,…,n), positive integers ki (i=1,2,…,n) and positive number a, where {u} denotes the fractional part of u. Furthermore, In,μα1,α2,…,αn(a) and k1,k2,…,knIn,μα1,α2,…,αn(a) can be transformed into the calculation of definite fractional part integral Iμα(a) and kIμα(a), which overcomes the shortcomings of low speed and poor accuracy of the multiple case. Moreover, some identities and recursive formulas of the above integrals are obtained.