A relationship between the algebraic multiplicity of eigenvalues of fractional integral operators and the simplicity of zeros of Mittag–Leffler functions

Temirkhan S. Aleroev; Yulong Li; Temirkhan S. Aleroev; Yulong Li

doi:10.3934/era.2026036

Electronic Research Archive

2026, Volume 34, Issue 2: 813-820. doi: 10.3934/era.2026036

Previous Article Next Article

Research article Special Issues

A relationship between the algebraic multiplicity of eigenvalues of fractional integral operators and the simplicity of zeros of Mittag–Leffler functions

Temirkhan S. Aleroev ¹,
Yulong Li ^{2
,
,}

1.
Department of Applied Mathematics, The National Research Moscow State University of Civil Engineering (NRU MGSU), Yaroslavskoe Highway 26, Moscow 129337, Russia
2.
Department of Mathematics, University of Dayton, 300 College Park Dayton, Dayton, OH 45469, USA

Received: 30 November 2025 Revised: 31 December 2025 Accepted: 05 January 2026 Published: 23 January 2026

In this short note, we establish a connection between the algebraic multiplicity of eigenvalues of compact integral operators and the simplicity of zeros of a class of the Mittag–Leffler functions $ E_{\alpha, \alpha}(z) $ and $ E_{\alpha, 2}(z) $ for $ 1 < \alpha < 2 $. Although these are two seemingly unrelated concepts in mathematics, our approach provides a functional‑analytic criterion for determining the simplicity of zeros in this class of Mittag–Leffler functions.
- algebraic simplicity,
- fractional Sturm–Liouville,
- Mittag–Leffler function,
- simplicity of zeros
Citation: Temirkhan S. Aleroev, Yulong Li. A relationship between the algebraic multiplicity of eigenvalues of fractional integral operators and the simplicity of zeros of Mittag–Leffler functions[J]. Electronic Research Archive, 2026, 34(2): 813-820. doi: 10.3934/era.2026036

Related Papers:

Abstract

In this short note, we establish a connection between the algebraic multiplicity of eigenvalues of compact integral operators and the simplicity of zeros of a class of the Mittag–Leffler functions $ E_{\alpha, \alpha}(z) $ and $ E_{\alpha, 2}(z) $ for $ 1 < \alpha < 2 $. Although these are two seemingly unrelated concepts in mathematics, our approach provides a functional‑analytic criterion for determining the simplicity of zeros in this class of Mittag–Leffler functions.

References

[1]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier Science B.V., Amsterdam, 2006.
[2]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993.
[3]	A. Y. Popov, A. M. Sedletskii, Distribution of roots of Mittag-Leffler functions, J. Math. Sci., 190 (2013), 209–409. https://doi.org/10.1007/s10958-013-1255-3 doi: 10.1007/s10958-013-1255-3
[4]	Y. Li, A. Telyakovskiy, E. Çelik, Analysis of one-sided 1-D fractional diffusion operator, Commun. Pure Appl. Anal., 21 (2022), 1673–1690. https://doi.org/10.3934/cpaa.2022039 doi: 10.3934/cpaa.2022039
[5]	A. Wiman, Über die Nullstellen der Funktionen $E_\alpha(x)$, Acta Math., 29 (1905), 217–234. https://doi.org/10.1007/BF02403204 doi: 10.1007/BF02403204
[6]	M. M. Dzhrbashyan, Integral Transforms and Representations of Functions in Complex Domains, (in Russian), Izdat. "Nauka", Moscow, 1966.
[7]	R. Gorenflo, A. A. Kilbas, F. Mainardi, S. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, 2nd edition, Springer Monographs in Mathematics, Springer, Berlin, 2020. https://doi.org/10.1007/978-3-662-61550-8
[8]	P. Van Mieghem, The Mittag-Leffler function, preprint, arXiv: 2005.13330.
[9]	T. S. Aleroev, M. Kirane, Y. F. Tang, Boundary value problems for differential equations of fractional order, J. Math. Sci., 194 (2013), 499–512. https://doi.org/10.1007/s10958-013-1543-y doi: 10.1007/s10958-013-1543-y
[10]	A. M. Sedletskii, Nonasymptotic properties of the roots of a function of Mittag-Leffler type, Math. Notes, 75 (2004), 372–386. https://doi.org/10.1023/B:MATN.0000023316.90489.fe doi: 10.1023/B:MATN.0000023316.90489.fe
[11]	T. S. Aleroev, Y. Li, The fractional Sturm-Liouville problem with the Caputo derivative may lose the principal eigenvalue, Fract. Calc. Appl. Anal., 28 (2025), 1706–1716. https://doi.org/10.1007/s13540-025-00429-x doi: 10.1007/s13540-025-00429-x
[12]	P. Eloe, Y. Li, On the first root of two-parametric Mittag–Leffler functions: A functional perspective, Integr. Transforms Special Funct., 36 (2025), 776–806. https://doi.org/10.1080/10652469.2025.2455502 doi: 10.1080/10652469.2025.2455502
[13]	R. Chaudhary, K. Diethelm, S. Hashemishahraki, On the separation of solutions to fractional differential equations of order $\alpha \in (1, 2)$, Appl. Numer. Math., 203 (2024), 84–96. https://doi.org/10.1016/j.apnum.2024.05.020 doi: 10.1016/j.apnum.2024.05.020

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)