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On the generalized Mittag-Leffler type functions. (English) Zbl 0935.33012
The authors study the generalized Mittag-Leffler type function $E_{\alpha,m,l}(z)= \sum_{k=0}^\infty c_kz^k$ with $$c_k= \prod_{i=0}^{k-1} \Gamma[\alpha (im+l)+1]/ \Gamma[\alpha (im+l+1)+1]$$ for $k=0,1,2,\dots$, $\alpha>0$, $m>0$, $\alpha(im+l)\ne 0,-1,-2,\dots$ and the particular case of these functions for $\alpha= n\in\bbfN$. This functions were introduced by {\it A. A. Kilbas} and {\it M. Saigo} [Differ. Integral Equ. 8, No. 5, 993-1011 (1995; Zbl 0823.45002)]. The authors show that $E_{\alpha,m,l}(z)$ is an entire function and they find the order and type and some recurrence formulae for such a function. For the particular case $E_{n,m,l}(z)$ connections with hypergeometric functions are discussed and differentiation properties are proved.

33E20Functions defined by series and integrals
33C20Generalized hypergeometric series, ${}_pF_q$
30D15Special classes of entire functions; growth estimates
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