  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.
