Bernoulli polynomials in several variables and summation of monomials over lattice points of a rational parallelotope

The Bernoulli polynomials for natural $x$ were first considered by J. Bernoulli (1713) in connection with the problem of summation of the powers of consecutive positive integers. For arbitrary $x$ these polynomials were studied by L. Euler. The term "Bernoulli polynomials" was introduced by Raabe (J. L. Raabe, 1851). The Bernoulli numbers and polynomials are well studied, and are widely used in various fields of theoretical and applied mathematics.
The article is devoted to some generalizations of the Bernoulli numbers and polynomials to the case of several variables. The concept of Bernoulli numbers associated to a rational cone generated by vectors with integer coordinates is defined. Using the Bernoulli numbers, we introduce the Bernoulli polynomials of several variables. Next we construct a difference operator acting on functions defined in a rational cone, and by methods of the theory of generating functions we prove a multidimensional analogue of the main property, which is the fact that the Bernoulli polynomials satisfy a difference equation.
Also, we calculate the values of the integrals of the Bernoulli polynomials over shifts of the fundamental parallelotope, and for the sum of monomials over integer points of a rational parallelotope we find a multidimensional analogue of the Bernoulli formula, where the sum above is expressed in terms of the integral of the Bernoulli polynomial over a parallelotope with variable "top" vertex.