Two-dimensional Mittag-Leffler-Konhauser polynomials: $k$-fractional calculus and associated properties

In this study, we introduced and investigated new classes of $k$-Mittag-Leffler-Konhauser polynomials and bivariate $k$-Mittag-Leffler functions, which encompass several well-known 2D polynomials and Mittag-Leffler functions as special cases. We explored their key properties, including double $k$-Riemann-Liouville fractional calculus, double Laplace transforms, and $k$-fractional calculus operators. To further support the theoretical results, a new section presents numerical validation through simulation examples, illustrating the accuracy and practical applicability of the proposed formulas. We conclude by proposing several open questions to inspire further research and continued exploration in this area.