Short Note on Generalised Bivariate Mittag-Leffler-Type Functions

In this short note, we aim to announce our joint result with Karimov and Hasanov (On Generalized Mittag-Leffler-Type Functions of Two Variables. Preprint, 2023. https://www.authorea.com/doi/pdf/10.22541/au.167575527.77144915 ) on generalised bivariate Mittag-Leffler-type functions \({{D}_{1}}\left ( x,y \right ),\ldots ,{{D}_{5}}\left ( x,y \right )\) . Special attention we paid to functions \({{E}_{1}}\left ( x,y \right ),\ldots , {{E}_{10}}\left ( x,y \right )\) as limiting cases of the functions \({{D}_{1}}\left ( x,y \right ),\) \(\ldots ,{{D}_{5}}\left ( x,y \right )\) . Following Horn’s method, we determine all possible cases of the convergence region of the function \({{D}_{1}}\left ( x,y \right ).\) Further, for the function \({{D}_{1}}\left ( x,y \right )\) integral representations of the Euler-type have been presented. The one-dimensional and two-dimensional Laplace transforms of the function are also defined. We have constructed a system of partial differential equations that is linked to the function \({{D}_{1}}\left ( x,y \right )\) .