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On the generalized Mittag-Leffler type functions. (English) Zbl 0935.33012

The authors study the generalized Mittag-Leffler type function ${E}_{\alpha ,m,l}\left(z\right)={\sum }_{k=0}^{\infty }{c}_{k}{z}^{k}$ with

${c}_{k}=\prod _{i=0}^{k-1}{\Gamma }\left[\alpha \left(im+l\right)+1\right]/{\Gamma }\left[\alpha \left(im+l+1\right)+1\right]$

for $k=0,1,2,\cdots$, $\alpha >0$, $m>0$, $\alpha \left(im+l\right)\ne 0,-1,-2,\cdots$ and the particular case of these functions for $\alpha =n\in ℕ$. This functions were introduced by A. A. Kilbas and M. Saigo [Differ. Integral Equ. 8, No. 5, 993-1011 (1995; Zbl 0823.45002)]. The authors show that ${E}_{\alpha ,m,l}\left(z\right)$ is an entire function and they find the order and type and some recurrence formulae for such a function. For the particular case ${E}_{n,m,l}\left(z\right)$ connections with hypergeometric functions are discussed and differentiation properties are proved.

##### MSC:
 33E20 Functions defined by series and integrals 33C20 Generalized hypergeometric series, ${}_{p}{F}_{q}$ 30D15 Special classes of entire functions; growth estimates
##### Keywords:
Mittag-Leffler type function