[Paper Review] The Mittag-Leffler function

Piet Van Mieghem|arXiv (Cornell University)|May 27, 2020

Matrix Theory and AlgorithmsComputer Science31 references65 citations

TL;DR

The paper provides a self-contained review of the Mittag-Leffler function E_{a,b}(z), including new results, covering complex analysis, special values, differentiation, relations, and applications in fractional calculus.

ABSTRACT

We review the function theoretical properties of the Mittag-Leffler function $E_{a,b}\left( z ight) $ in a self-contained manner, but also add new results; more than half is new!

Motivation & Objective

• Delineate the functional properties of the Mittag-Leffler function E_{a,b}(z) and set a rigorous analytical framework.
• Present new results on its differentiation, recursion, and special values to enhance applicability in fractional calculus.
• Connect E_{a,b}(z) with related functions (e.g., hypergeometric, cosh, erfc) and derive useful identities.
• Discuss zeros, order, and growth, and establish fundamental relations that support probabilistic and physical applications.

Proposed method

• Define and analyze E_{a,b}(z) = sum_{k=0}^{∞} z^{k} / Γ(b + a k).
• Derive key identities such as the differentiation rule a dz/dz E_{a,b}(z) = E_{a,b-1}(z) − (b-1) E_{a,b}(z).
• Obtain and use recursion and shift formulas like E_{a,b}(z) = (1/z)(E_{a,b−a}(z) − 1/Γ(b−a)).
• Explore special values (e.g., E_{1,1}(z)=e^{z}, E_{2,1}(z)=cosh(√z)) and cyclotomic decomposition (E_{am,b}(z^{m}) relations).
• Analyze order and growth, zeros behavior, and Hadamard-type bounds for Γ(·) to bound E_{a,b}(z).
• Develop differentiation recursions and expansions for general and fractional a, including E_{1/n,b}(z^{1/n}) cases.

Experimental results

Research questions

• RQ1 What are the fundamental analytic properties (entirety, order) of E_{a,b}(z) for Re(a)>0?
• RQ2 How can one differentiate and relate E_{a,b}(z) across shifts in b and a, and what are the resulting recursive structures?
• RQ3 What special values and representations connect E_{a,b}(z) to elementary or classical special functions?
• RQ4 How do cyclotomic and multi-valued decompositions of E_{a,b}(z) behave and what are their implications?
• RQ5 What bounds, zeros, and growth results can be established to aid applications in fractional calculus and related fields?

Key findings

• E_{a,b}(z) is an entire function of order ρ = 1/a for Re(a) > 0 and any b.
• A fundamental differentiation rule is az d/dz E_{a,b}(z) = E_{a,b-1}(z) − (b−1) E_{a,b}(z).
• Several shift/recurrence formulas hold, including E_{a,b}(z) = (1/z)(E_{a,b−a}(z) − 1/Γ(b−a)) and E_{a,b}(z) = (1/Γ(b)) + z E_{a,b+a}(z).
• Cyclotomic property: E_{am,b}(z^{m}) = (1/m) ∑_{r=0}^{m−1} E_{a,b}(z e^{i 2π r/m}).
• Special cases yield elementary functions (e.g., E_{1,1}(z)=e^{z}, E_{2,1}(z)=cosh(√z)) and connections to hypergeometric functions (E_{1,b}(z)=M(1,b,z)/Γ(b)).
• Bounds and asymptotics include Hadamard-type bounds and comparisons with e^{x^{1/a}} providing sharp growth estimates.
• A fractional-a analysis yields representations for E_{1/n,b}(x) in terms of incomplete gamma functions and exponentials, highlighting the order n of these functions.
• The logarithmic derivative d/dz log E_{a,b}(z) is positive for z>0 and b>1, indicating monotonic growth of log E_{a,b}(z) on the real axis.
• Taylor expansions around arbitrary z0 are possible, with a special differentiated recursion enabling expansions via E_{a,b−j}(z0).