Schlomilch and Bell Series for Bessel's Functions, with Probabilistic Applications

TL;DR

This paper introduces new series for Bessel functions and explores their properties, connecting them with Stirling and Bell numbers, while also applying these findings to improve inequalities and numerical methods in probability theory.

Contribution

It presents novel series for Bessel functions and applies them to derive improved probabilistic inequalities and numerical algorithms, linking special functions with probability bounds.

Findings

Derived exact constants in moment inequalities

Improved asymptotic properties of inequalities

Developed a numerical algorithm for calculations

Abstract

We have introduced and investigated so-called Shlomilchs and Bells series for modified Bessel's functions, namely, their asymptotic and non-asymptotic properties, connection with Stirling's and Bell's numbers etc. We have obtained exact constants in the moment inequalities for sums of centered independent random variables, improved their asymptotical properties, found lower and upper bounds, calculated a more exact approximation, elaborated the numerical algorithm for their calculation, studied the class of smoothing, etc.