Self-Similar Exponential Approximants

An approach is suggested defining effective sums of divergent series in the form of self-similar exponential approximants. The procedure of constructing these approximants from divergent series with arbitrary noninteger powers is developed. The basis of this construction is the self-similar approximation theory. Control functions governing the convergence of exponentially renormalized series are defined from stability and fixed-point conditions and from additional asymptotic conditions when the latter are available. The stability of the calculational procedure is checked by analyzing cascade multipliers. A number of physical examples for different statistical systems illustrate the generality and high accuracy of the approach.