On the first root of two-parametric Mittag–Leffler functions: a functional perspective
Paul W. Eloe, Yulong Li
Abstract
We construct a novel class of integral operators whose eigenvalue reciprocals correspond to the roots (zeros) of two-parametric Mittag–Leffler functions Eα,α∗(z). Using the theory of μ0-positive operators, we demonstrate that the smallest roots (in modulus) of Eα,α∗(z) must be real. Moreover, by employing the relationship between roots of Mittag–Leffler functions Eα,α∗(z) and the eigenvalues of the integral spectral problem, together with new techniques involving fractional integral operators and fractional Sobolev spaces, we establish the existence, non-existence, and simplicity of the real root of Eα,α∗(−z) for a large class of parameters α,α∗: 1≤α<∞,0<α∗<α+ϵ. These findings extend the theory previously developed for the case 0<α<1 by other researchers, such as A. M. Sedletskiĭ in 2004.