MMN-3963

In this paper, we first introduce new class of 2D-$k$-Laguerre-Konhauser polynomials, $_{\delta }L_{k,n}^{(\alpha ,\beta )}(x,y)$, which generalizes the 2D-Laguerre-Konhauser polynomials (see \cite{26a}). Then, we define a new family of bivariate \textit{k}-Mittag-Leffler functions $E_{k,\alpha ,\beta ,\delta }^{(\gamma )}(x,y)$ and establish the \textit{k}-Riemann-Liouville double fractional integral and derivative of the functions $E_{k,\alpha ,\beta ,\delta }^{(\gamma )}(x,y)$. Moreover, we introduce an integral operator $_{k}\varepsilon _{\alpha ,\beta ,\delta ;\omega _{1},\omega _{2};a^{+},c^{+}}^{(\gamma )}$ which contains the bivariate $k$-Mittag-Leffler functions $E_{k,\alpha ,\beta ,\delta }^{(\gamma )}(x,y)$ in the kernel and investigate the semigroup property of this operator. Finally, the left inverse operator of the integral operator $_{k}\varepsilon _{\alpha ,\beta ,\delta ;\omega _{1},\omega _{2};a^{+},c^{+}}^{(\gamma )}$ is constructed.

Vol. 24 (2023), No. 2, pp. 861-876

DOI: 10.18514/MMN.2023.3963

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