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

Maged G. Bin-Saad; Mohra Zayed; Ali Z. Bin-Alhage; Maged G. Bin-Saad; Mohra Zayed; Ali Z. Bin-Alhage

doi:10.3934/era.2026006

Electronic Research Archive

2026, Volume 34, Issue 1: 90-112. doi: 10.3934/era.2026006

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Research article

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

1.
Department of Mathematics, College of Education, University of Aden, Aden, Yemen
2.
Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia

Received: 25 October 2025 Revised: 06 December 2025 Accepted: 11 December 2025 Published: 30 December 2025

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.
- Mittag-Leffler function,
- Konhauser polynomials,
- $ k $-fractional integral and derivative,
- double Laplace transform,
- convolution integral equation
Citation: Maged G. Bin-Saad, Mohra Zayed, Ali Z. Bin-Alhage. Two-dimensional Mittag-Leffler-Konhauser polynomials: $ k $-fractional calculus and associated properties[J]. Electronic Research Archive, 2026, 34(1): 90-112. doi: 10.3934/era.2026006

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Abstract

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.

References

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