Conversions of the convolution operation to the sum and the asymptotic behavior of the stable polynomials coefficients

Consider the integral transformations which convert convolution in the domain of originals (functions of scalar real variable) into the sum of images (functions of scalar real variable). All these transformations are given up to a linear operator.
We discuss the properties of one of these transformations, which converts any exponent the exponent: its relationship with the Laplace transform, transform of some particular functions and operations differentiation, integration, shift, time scaling, multiplication by the exponent, etc.
Transformations of this type we call cumulative by analogy with the transition from the density distribution of a random variable to its cumulants. We show that Newton’s formulas that realize the relation of sums of the same powers of the roots of a polynomial with its coefficients are cumulative transformation. Also, any transition of real variable function to its phase (same as the logarithm of the module of its Fourier transform) is.
We discuss the possible applications and obtain the conditions under which the sequence of coefficients of a stable polynomial with increasing its degree is asymptotically normal.