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\begin{document}

\title{ON GENERALIZED MITTAG-LEFFLER-TYPE FUNCTIONS OF TWO VARIABLES}



\author[1]{ERKINJON KARIMOV}%
\author[1]{Anvar Hasanov}%
\affil[1]{Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan}%


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  \date{May 24, 2024}


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\begin{abstract}
We aim to study Mittag-Leffler type functions of two variables D 1 ( x ,
y ) , . . . , D 5 ( x , y ) by analogy with the Appell hypergeometric
functions of two variables,. Moreover, we targeted functions E 1 ( x , y
) , . . . , E 10 ( x , y ) as limiting cases of the functions D 1 ( x ,
y ) , . . . , D 5 ( x , y ) and studied certain properties, as well.
Following Horn's method, we determine all possible cases of the
convergence region of the function D 1 ( x , y ) . Further, for a
generalized hypergeometric function D 1 ( x , y ) (Mittag-Leffler type
function) integral representations of the Euler type are proved.
One-dimensional and two-dimensional Laplace transforms of the function
are also defined. We have constructed a system of partial differential
equations which is linked with the function D 1 ( x , y ) .%
\end{abstract}%



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