Comments on the properties of Mittag-Leffler function

Abstract

The properties of Mittag-Leffler function is reviewed within the framework of an umbral formalism. We take advantage from the formal equivalence with the exponential function to define the relevant semigroup properties. We analyse the relevant role in the solution of Schrödinger type and heat-type fractional partial differential equations and explore the problem of operatorial ordering finding appropriate rules when non-commuting operators are involved. We discuss the coherent states associated with the fractional Schödinger equation, analyze the relevant Poisson type probability amplitude and compare with analogous results already obtained in the literature.

Operator Ordering and Solution of Pseudo-Evolutionary Equations

“…by using the formal multiplication operatorx, this expression is also valid when part of larger expressions. 2 Here and throughout this paper, in expressions such as ϕ(x + κλ + 2λ∂ x )1, the occurrence of the symbol "1" entails that the expression is to be evaluated by expanding ϕ(x + κλ + 2λ∂ x ) into normal-ordered form (i.e. into a series in the normal-ordered monomialsx r ∂ s x for r, s ≥ 0), followed by acting on 1 (which due to ∂ s x 1 = 0 for s > 0 in effect amounts to dropping all terms of the expansion involving non-zero powers of ∂ x ).…”

“…This permits us to determine the solution of the fractional pseudo-evolution equation (49) in closed form as 4 Here, the last term in (54) arises due to the action of the fractional derivative in the sense of Riemann-Liouville onto the constant term 1 of E α,β (Mt µ ), i.e. it is the contribution ∂ µ t 1 = t −µ /Γ(1 − µ) (compare [2]).…”

“…It is evident that the strategy we have in mind is that of treating E as a kind of exponential function (the eigenfunction of the ordinary derivative operator) -in this way, we can recover most of the techniques associated with operator disentanglement and time-ordering (whenever necessary). A first study in this direction has been addressed by D. Babusci and one of the present authors in [1], which has later been specialized in a number of papers for situations in whichD t is a fractional [2] or Laguerre [3] derivative. In elaborating this program, we have been faced with some difficulties associated with the fact that the corresponding eigenfunctions E lack the semi-group property, namely…”

Mittag-Leffler function and fractional differential equations

“…For the remain value of α this relation is violate. For α = 1 the convolution property of two ML functions has the form [9,11]…”

“…It is obtained by employing the so-called shift (umbra) representation of ML function. The umbral approach is frequently applied in the characterization of special functions and polynomials [9,16,42] especially in the combinatorial description of the latter one [16,44,49]. Usually, in many cases the use of umbral operational technique notably simplifies the calculations like, e.g., integrations and summations [2,14,42].…”