The mean-value theorem for trigonometric sums on the sequence of binomial type polynomials

Abstract

The mean-value theorem for trigonometric sums on the sequence of binomial type polynomials was proved.
As known, the classical I. M. Vinogradov mean-value theorem belong to the sequence of polynomials of the form $\{x^n, n\geq 0\}.$ Estimates of sums of the kind
$$
\sum_{m\leq P}e^{2\pi if(m)}, f(m)=\sum_{k=0}^n\alpha_kp_k(m),
$$
are the important application of the finding mean-value theorem.
Here $p_k(x)$ is the sequence integer-valued polynomials of the binomial type, but a set of numbers $(\alpha_1\alpha_1,\dots,\alpha_n)$ represents a point of the $n$-fold unit cube $\Omega: 0\leq \alpha_1,\dots,\alpha_n<1.$