New estimate for the multinomial Mittag-Leffler function
aa r X i v : . [ m a t h . G M ] J un NEW ESTIMATE FOR THE MULTINOMIALMITTAG-LEFFLER FUNCTIONMurat Mamchuev Abstract
In this paper, a new estimate is obtained for the multinomial Mittag-Leffler function. This function was introduced by Yuri Luchko and RudolfoGorenflo as the fundamental solution of the ordinary differential equationof fractional discrete distributed order.
Keywords: multinomial Mittag-Leffler function, fractional differential equa-tions, esitimate.
The multinomial Mittag-Leffler function is defined as [1] E ( µ ,µ ,...,µ n ) ,γ ( z , z , ..., z n )= ∞ X k =0 X l + ... + l n = k ( k ; l , ..., l n ) z l · ... · z l n Γ( γ + P ni =1 µ i l i ) , here ( k ; l , ..., l n ) denotes the multinomial coefficient( k ; l , ..., l n ) = k ! l ! · ... · l n ! , k = n X i =1 l i , where l i ( i = 1 , ..., n ) are non-negative integers.We also need the following definition of the Mittag-Leffler-type function [2] E µ,ν ( z ) = ∞ X k =0 z k Γ( µk + ν ) . Lemma 1.
Let < µ < µ < ... < µ n , γ > , then the estimate (cid:12)(cid:12) E ( µ ,µ ,...,µ n ) ,γ ( z , z , ..., z n ) (cid:12)(cid:12) ≤ CE µ ,γ ( | z | + | z | + ... + | z n | ) holds, here C = 1 + Γ( γ + µ n )Γ( γ ) , n ∈ N is a number such that µ n < x − γ < µ ( n + 1) , x = min x> Γ( x ) . Proof.
The following relation X l + ... + l n = k ( k ; l , ..., l n ) n Y i =1 z l i i = ( z + ... + z n ) k , (1) Institute of Applied Mathematics and Automation of KBSC of RAS, Shortanova str.89-A, Nalchik, 360000, Kabardino-Balkar Republic, Russia, E-mail: [email protected] γ + n X i =1 µ i l i = γ + µ j n X i =1 l i + n X i =1 ( µ i − µ j ) l i = γ + µ j k + n X i =1 ( µ i − µ j ) l i , ≤ j ≤ n. Consequently, γ + µ k < γ + n X i =1 µ i l i < γ + µ n k. Sinse the function x ) has only one maximum on the interval (0 , ∞ ) whichreached at the point x = 1 , ..., and decreases on the interval ( x , ∞ ) , then there exists the number n ∈ N such that the inequality1Γ( γ + P ni =1 µ i l i ) < γ + µ k ) (2)holds for every numbers k > n . For 0 ≤ k ≤ n we have the inequality1Γ( γ + P ni =1 µ i l i ) < C γ + µ k ) , (3)where C = max ≤ k ≤ n Γ( γ + µ k )min ≤ k ≤ n Γ( γ + P ni =1 µ i l i ) = Γ( γ + µ n )Γ( γ ) > . Hence, by virtue of (1) – (3), we get (cid:12)(cid:12) E ( µ ,µ ,...,µ n ) ,γ ( z , z , ..., z n ) (cid:12)(cid:12) = ∞ X k =0 X l + ... + l n = k ( k ; l , ..., l n ) | z | l · ... · | z n | l Γ( γ + P ni =1 µ i l i ) ≤ C ∞ X k =0 γ + µ k ) X l + ... + l n = k ( k ; l , ..., l n ) n Y i =1 | z i | l i = C ∞ X k =0 ( | z | + ... + | z n | ) k Γ( γ + µ k ) = CE µ ,γ ( | z | + | z | + ... + | z n | ) , where C = 1 + C . The Lemma 1 is proved.
References [1] Y. Luchko, R. Gorenflo, An operational method for solving fractional dif-ferential equations with the Caputo derivatives, Acta Math. Vietnam 24(1999) 207-233.[2] M.M. Dzherbashyan,