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

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