In this paper, the following multiple fractional part integrals I n , β α 1 , α 2 , ? , α n = ? 0 , 1 n ? j = 1 n x j α j { S n - 1 } β d x 1 ? d x n and J n , β α = ? 0 , 1 n S n α { S n - 1 } β d x 1 ? d x n are studied for positive integer n and complex values of α , β , α j ( j = 1 , 2 , ? , n ) , where { u } denotes the fractional part of u , R ( s ) denotes the real part of s and S n = x 1 + x 2 + ? + x n . It is proved that I 1 , β α can be represented as a linear combination of the Riemann zeta function, the Beta function and Euler's constant as R ( β ) - 1 . Moreover, I n , β α 1 , α 2 , ? , α n can be expressed by I n - 1 , β α 1 , α 2 , ? , α n - 1 , the Beta function and the incomplete Beta function for n = 2 , 3 . In addition, the recurrence formula of J n , β α ( n = 2 , 3 , ? ) is established and J n , β α can be expressed by I 1 , β α , logarithmic function and some binomial coefficients.
['Zhongfeng Sun', 'Aijuan Li', 'Huizeng Qin']
Further results on generalized multiple fractional part integrals for complex values