Finite and infinite order differential properties of the reduced Mittag--Leffler polynomials

TL;DR

This work reframes the Mittag–Leffler polynomials as real polynomials with favorable properties, introducing the reduced sequence $\{\varphi_n(x)\}$ obtained from $g_n(x)$ via $\varphi_n(x)=\frac{g_{n+1}(ix)}{i^{n+1}x}$. It derives a real-orthogonal framework, a three-term recurrence, and explicit generating and Fourier-transform formulas for the reduced polynomials, and develops novel differential properties, including finite and infinite order differential equations and an eigenoperator formulation $\big(\cos D+x\sin D-(n+1)\big)\hat{\varphi}_n(x)=0$. The results reveal a rich structure: a Turán-type inequality, explicit derivative relations, and a Boas–Buck-type quasi-monomial representation with lowering operator $\mathcal{L}_x=2\tan(D_x/2)$, linking the reduced Mittag–Leffler polynomials to classical families such as Meixner–Pollaczek. These contributions provide a robust real-polynomial framework and operational tools for further spectral and orthogonality analyses of the family.

Abstract

This paper deals with the Mittag-Leffler polynomials (MLP) by extracting their essence which consists of real polynomials with fine properties. They are orthogonal on the real line instead of the imaginary axes for MLP. Beside recurrence relations and zeros, we will point to the closed form of its Fourier transform. The most important contribution consists of the new differential properties, especially the finite and infinite differential equation.

Theorems & Definitions (14)

• Theorem 2.1
• Remark 3.1
• Example 3.1
• Theorem 3.1
• Theorem 4.1
• Remark 4.1
• Theorem 5.1
• Corollary 5.1
• Theorem 5.2
• Example 5.1