Mean-Value Theorem for Multiple Trigonometric Sums on the Sequence of Bell Polynomials
Informatics and Automation, Analytic and Combinatorial Number Theory, Tome 314 (2021), pp. 301-310
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A mean-value theorem for multiple trigonometric (exponential) sums on the sequence of Bell polynomials is proved. It generalizes I. M. Vinogradov's and G. I. Arkhipov's theorems. As is well known, a mean-value theorem of this type is at the core of Vinogradov's method. The Bell polynomials are very closely related to the Faà di Bruno theorem on higher order derivatives of a composite function. As an application of the mean-value theorem proved in the paper, estimates for the sums $\sum _{n_1\leq P}\dots \sum _{n_r\leq P}e^{2\pi i(\alpha _1Y_1(n_1)+\dots +\alpha _rY_r(n_1,\dots ,n_r))}$ are obtained, where $\alpha _s$ are real numbers and $Y_s(n_1,\dots ,n_s)$ are the degree $s$ Bell polynomials, $1\leq s\leq r$.
Keywords:
mean-value theorems of Vinogradov and Arkhipov, sequence of Bell polynomials, Faà di Bruno theorem.
@article{TRSPY_2021_314_a13,
author = {V. N. Chubarikov},
title = {Mean-Value {Theorem} for {Multiple} {Trigonometric} {Sums} on the {Sequence} of {Bell} {Polynomials}},
journal = {Informatics and Automation},
pages = {301--310},
publisher = {mathdoc},
volume = {314},
year = {2021},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TRSPY_2021_314_a13/}
}
V. N. Chubarikov. Mean-Value Theorem for Multiple Trigonometric Sums on the Sequence of Bell Polynomials. Informatics and Automation, Analytic and Combinatorial Number Theory, Tome 314 (2021), pp. 301-310. http://geodesic.mathdoc.fr/item/TRSPY_2021_314_a13/