Characterization of finiteness for the fractional moment-generating function

Establish a sharper characterization of when the L-fractional moment-generating function of a random variable is finite, potentially via the Cauchy–Hadamard theorem, beyond the current ratio-test-based criteria.

Background

The paper introduces a fractional moment-generating function tied to Ea and proves partial finiteness results via the ratio test. A more complete characterization would parallel classical moment-generating function theory.