New estimate for the multinomial Mittag-Leffler function

TL;DR

This paper introduces a new estimate for the multinomial Mittag-Leffler function, which is fundamental in fractional differential equations, enhancing understanding and potential applications in this mathematical area.

Contribution

The paper provides a novel estimate for the multinomial Mittag-Leffler function, advancing the analytical tools available for fractional differential equations.

Findings

New estimate improves understanding of the multinomial Mittag-Leffler function

Enhances analytical methods for fractional differential equations

Potential applications in mathematical modeling of complex systems

Abstract

In this paper, a new estimate is obtained for the multinomial Mittag-Leffler function. This function was introduced by Yuri Luchko and Rudolfo Gorenflo as the fundamental solution of the ordinary differential equation of fractional discrete distributed order.

Equations

$$E_{({\mu }_1,{\mu }_2,...,{\mu }_n),\gamma }(z_1,z_2,...,z_n)$$

$$=\sum _{k=0}^{\infty }\sum _{l_1+...+l_n=k}(k;l_1,...,l_n)\frac{z_1^{l_1}\cdot ...\cdot z_n^{l_1}}{\Gamma (\gamma +{\sum }_{i=1}^n{\mu }_i{l}_i)},$$

$$(k;l_1,...,l_n)=\frac{k!}{l_1!\cdot ...\cdot l_n!},k=\sum _{i=1}^n{l}_i,$$

$$E_{\mu ,\nu }(z)=\sum _{k=0}^{\infty }\frac{z^k}{\Gamma (\mu k+\nu )}.$$

$$\left|E_{({\mu }_1,{\mu }_2,...,{\mu }_n),\gamma }(z_1,z_2,...,z_n)\right|\le C{E}_{{\mu }_1,\gamma }(|z_1|+|z_2|+...+|z_n|)$$

$${\mu }_1{n}_0<x_0-\gamma <{\mu }_1(n_0+1),x_0=\underset{x>0}{\min }\Gamma (x).$$

$$\sum _{l_1+...+l_n=k}(k;l_1,...,l_n)\prod _{i=1}^n{z}_i^{l_i}=(z_1+...+z_n{)}^k,$$

$$\gamma +\sum _{i=1}^n{\mu }_i{l}_i=\gamma +{\mu }_j\sum _{i=1}^n{l}_i+\sum _{i=1}^n({\mu }_i-{\mu }_j)l_i=\gamma +{\mu }_{j}k+\sum _{i=1}^n({\mu }_i-{\mu }_j)l_i,0\le j\le n.$$

$$\gamma +{\mu }_{1}k<\gamma +\sum _{i=1}^n{\mu }_i{l}_i<\gamma +{\mu }_{n}k.$$

$$\frac{1}{\Gamma (\gamma +{\sum }_{i=1}^n{\mu }_i{l}_i)}<\frac{1}{\Gamma (\gamma +{\mu }_{1}k)}$$

$$\frac{1}{\Gamma (\gamma +{\sum }_{i=1}^n{\mu }_i{l}_i)}<C_0\frac{1}{\Gamma (\gamma +{\mu }_{1}k)},$$

$$C_0=\frac{\underset{0\le k\le n_0}{\max }\Gamma (\gamma +{\mu }_{1}k)}{\underset{0\le k\le n_0}{\min }\Gamma (\gamma +{\sum }_{i=1}^n{\mu }_i{l}_i)}=\frac{\Gamma (\gamma +{\mu }_1{n}_0)}{\Gamma (\gamma )}>0.$$

$$\left|E_{({\mu }_1,{\mu }_2,...,{\mu }_n),\gamma }(z_1,z_2,...,z_n)\right|$$

$$=\sum _{k=0}^{\infty }\sum _{l_1+...+l_n=k}(k;l_1,...,l_n)\frac{|z_1{|}^{l_1}\cdot ...\cdot |z_n{|}^{l_1}}{\Gamma (\gamma +{\sum }_{i=1}^n{\mu }_i{l}_i)}$$

$$\le C\sum _{k=0}^{\infty }\frac{1}{\Gamma (\gamma +{\mu }_{1}k)}\sum _{l_1+...+l_n=k}(k;l_1,...,l_n)\prod _{i=1}^n|z_i{|}^{l_i}$$

$$=C\sum _{k=0}^{\infty }\frac{(|z_1|+...+|z_n|{)}^k}{\Gamma (\gamma +{\mu }_{1}k)}=C{E}_{{\mu }_1,\gamma }(|z_1|+|z_2|+...+|z_n|),$$

Full text

Keywords: multinomial Mittag-Leffler function, fractional differential equations, esitimate.

The multinomial Mittag-Leffler function is defined as [1]

[TABLE]

[TABLE]

here $(k;l_{1},...,l_{n})$ denotes the multinomial coefficient

[TABLE]

where $l_{i}(i=1,...,n)$ are non-negative integers.

We also need the following definition of the Mittag-Leffler-type function [2]

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Lemma 1. Let $0<\mu_{1}<\mu_{2}<...<\mu_{n},$ $\gamma>0,$ then the estimate

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holds, here $C=1+\frac{\Gamma(\gamma+\mu_{1}n_{0})}{\Gamma(\gamma)},$ $n_{0}\in{\mathbb{N}}$ is a number such that

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Proof. The following relation

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can be proved by the mathematical induction method.

Note that

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Consequently,

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Sinse the function $\frac{1}{\Gamma(x)}$ has only one maximum on the interval $(0,\infty)$ which reached at the point $x_{0}=1,4616321...,$ and decreases on the interval $(x_{0},\infty),$ then there exists the number $n_{0}\in{\mathbb{N}}$ such that the inequality

[TABLE]

holds for every numbers $k>n_{0}.$ For $0\leq k\leq n_{0}$ we have the inequality

[TABLE]

where

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Hence, by virtue of (1) – (3), we get

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[TABLE]

[TABLE]

[TABLE]

where $C=1+C_{0}.$

The Lemma 1 is proved.