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Fractional Slice Regular Functions of a Quaternionic Variable

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Abstract

The theory of slice regular functions of a quaternionic variable on the unit ball of the quaternions was introduced by Gentili and Struppa in 2006 and nowadays it is a well established function theory, especially in view of its applications to operator theory. In this paper, we introduce the notion of fractional slice regular functions of a quaternionic variable defined as null-solutions of a fractional Cauchy–Riemann operators. We present a fractional Cauchy–Riemann operator in the sense of Riemann–Liouville and then in the sense of Caputo, with orders associated to an element of \((0,1)\times {\mathbb {R}} \times (0,1)\times {\mathbb {R}}\) for some axially symmetric slice domains which are new in the literature. We prove a version of the representation theorem, of the splitting lemma and we discuss a series expansion.

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References

  1. Alpay, D., Colombo, F., Sabadini, I.: Slice hyperholomorphic Schur analysis. Oper. Theory Adv. Appl. 256 (2016)

  2. Baleanu, D., Restrepo, J.E., Suragan, D.: A class of time-fractional Dirac type operators. Chaos Solitons Fractals 143, 15 (2021)

    Article  MathSciNet  Google Scholar 

  3. Bernstein, S.: A fractional Dirac operator. Oper. Theory Adv. Appl. 252, 27–41 (2016)

    MathSciNet  Google Scholar 

  4. Cerroni, C.: From the theory of congeneric surd equations to Segre’s bicomplex numbers. Hist. Math. 44(3), 232–251 (2017)

    Article  MathSciNet  Google Scholar 

  5. Coloma, N., Di Teodoro, A., Ochoa-Tocachi, D., et al.: Fractional elementary bicomplex functions in the Riemann–Liouville sense. Adv. Appl. Clifford Algebras 31(4), 29–63 (2021)

    Article  MathSciNet  Google Scholar 

  6. Colombo, F., Gantner, J.: Quaternionic closed operators, fractional powers and fractional diffusion processes. Oper. Theory Adv. Appl. 274(2), 1045–1100 (2016)

    Google Scholar 

  7. Colombo, F., Gantner, J.: Fractional powers of quaternionic operators and Kato’s formula using slice hyperholomorphicity. Trans. Am. Math. Soc. 370(2), 1045–1100 (2018)

    Article  MathSciNet  Google Scholar 

  8. Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative Functional Calculus. Theory and Applications of Slice Hyperholomorphic Functions. Birkhauser, Basel (2011)

    Book  Google Scholar 

  9. Colombo, F., Sabadini, I., Struppa, D.C.: Entire Slice Regular Functions. Springer Briefs in Mathematics. Springer, Boston (2016)

    Book  Google Scholar 

  10. Contharteze Grigoletto, E., Capelas de Oliveira, E.: Fractional versions of the fundamental theorem of calculus. Appl. Math. 4(3), 23–33 (2013)

    Article  Google Scholar 

  11. Delgado, B., Macías-Díaz, J.: On the general solutions of some non-homogeneous div-curl systems with Riemann–Liouville and Caputo fractional derivatives. Fractal Fract. 3(3), 1045–1100 (2021)

    Google Scholar 

  12. Dou, X., Jin, M., Ren, G., et al.: A new approach to slice analysis via slice topology. Adv. Appl. Clifford Algebras 31(5), 1045–1100 (2021)

    Article  MathSciNet  Google Scholar 

  13. Dou, X., Ren, G., Sabadini, I.: Extension theorem and representation formula in non-axially symmetric domains for slice regular functions. J. Eur. Math. Soc. 31(5), 1045–1100 (2021)

    Google Scholar 

  14. Ferreira, M., Vieira, N.: Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann–Liouville case. Complex Anal. Oper. Theory 10(5), 1081–1100 (2016)

    Article  MathSciNet  Google Scholar 

  15. Ferreira, M., Vieira, N.: Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives. Complex Var. Elliptic Equ. 62(9), 1237–1253 (2017)

    Article  MathSciNet  Google Scholar 

  16. Ferreira, M., Krausshar, R.S., Rodrigues, M.M., et al.: A higher dimensional fractional Borel–Pompeiu formula and a related hypercomplex fractional operator calculus. Math. Methods Appl. Sci. 42(10), 3633–3653 (2019)

    Article  MathSciNet  Google Scholar 

  17. Gentili, G., Stoppato, C.: A local representation formula for quaternionic slice regular functions. Proc. Am. Math. Soc. 149(5), 2025–2034 (2021)

    Article  MathSciNet  Google Scholar 

  18. Gentili, G., Struppa, D.C.: A new theory of regular function of a quaternionic variable. Adv. Math. 216(9), 279–301 (2007)

    Article  MathSciNet  Google Scholar 

  19. Gentili, G., Stoppato, C., Struppa, D.C.: Regular Functions of a Quaternionic Variable. Springer Monographs in Mathematics, vol. 149(5), pp. 2025–2034. Springer (2013)

  20. González-Cervantes, J.O., Bory-Reyes, J.: A quaternionic fractional Borel–Pompeiu type formula. Fractal 30(1), 2025–2034 (2022)

    Article  Google Scholar 

  21. González-Cervantes, J.O., Bory-Reyes, J.: A bicomplex \((\vartheta ,\varphi )-\)weighted fractional Borel–Pompeiu type formula. J. Math. Anal. Appl. 520(2), 2025–2034 (2023)

    Article  MathSciNet  Google Scholar 

  22. Kähler, U., Vieira, N.: Fractional Clifford analysis. In: Hypercomplex Analysis: New Perspectives and Applications. Trends in Mathematics, vol. 520(2), pp. 191–201 (2014)

  23. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Hypercomplex Analysis: New Perspectives and Applications. Trends in Mathematics. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

  24. Leibniz, G.W.: TMathematische Schriften: aus den Handschriften der Königlichen Bibliothek zu Hannover. Briefwechsel zwischen Leibniz, Wallis, Varignon, Guido Grandi, Zendrini, Hermann und Freiherrn von Tschirnhaus, vol. 1. Druck und Verlag von H.W. Schmidt. Halle, Germany (1859)

  25. Liouville, J.: Mémoire sur le calcul des différentielles a indices quelconques. J. Ecole Polytech. 13(2), 71–162 (1832)

    Google Scholar 

  26. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    Google Scholar 

  27. Oldham, K.B., Spanier, J.: The Fractional Calculus. Dover Publications, New York (2006)

    Google Scholar 

  28. Ortigueira, M.D.: Fractional Calculus for Scientists and Engineers. Lecture Notes in Electrical Engineering, vol. 84. Springer, Dordrecht (2011)

    Book  Google Scholar 

  29. Peña Pérez, Y., Abreu Blaya, R., Árciga Alejandre, M.P., et al.: Biquaternionic reformulation of a fractional monochromatic Maxwell system. Adv. High Energy Phys. 13(2), 71–162 (2020)

    MathSciNet  Google Scholar 

  30. Podlubny, I.: Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999)

  31. Riemann, B.: Versuch einer allgemeinen Auffassung der Integration und Differentiation (An attempt to a general understanding of integration and differentiation) (1847) In: Weber, H. (ed.), Bernhard Riemanns gesammelte mathematische Werke und wissenschaftlicher Nachlass. Dover Publications, Germany (1953)

  32. Ross, B.: A brief history and exposition of the fundamental theory of fractional calculus. In: Ross, B. (ed.) Fractional Calculus and Its Applications. Lecture Notes in Mathematics, vol. 457. Springer, Berlin (1975)

    Chapter  Google Scholar 

  33. Samko, S., Kilbas, A.A., Marichev, O.: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach Science Publlication, London (1993)

    Google Scholar 

  34. Tarasov, V.E.: No violation of the Leibniz rule. No VTRMB fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 18(11), 2945–2948 (2013)

    Article  MathSciNet  Google Scholar 

  35. Valério, D., Trujillo, J.J., Rivero, M., et al.: Fractional calculus: a survey of useful formulas. Eur. Phys. J. Spec. Top. 222(11), 1827–1846 (2013)

    Article  Google Scholar 

  36. Vieira, N.: Fischer decomposition and Cauchy–Kovalevskaya extension in fractional Clifford analysis: the Riemann–Liouville case. Proc. Edinb. Math. Soc. II 60(1), 251–272 (2017)

    Article  MathSciNet  Google Scholar 

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Funding

This work was partially supported by Instituto Politécnico Nacional (Grant Numbers SIP20232103, SIP20230312) and CONACYT.

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Authors and Affiliations

  1. Departamento de Matemáticas, ESFM-Instituto Politécnico Nacional, Av. Instituto Politécnico Nacional, 07338, Mexico City, CDMX, Mexico

    José Oscar González-Cervantes

  2. SEPI, ESIME-Zacatenco, Instituto Politécnico Nacional, Av. Instituto Politécnico Nacional, 07338, Mexico City, CDMX, Mexico

    Juan Bory-Reyes

  3. Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi 9, 20133, Milan, Italy

    Irene Sabadini

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  1. José Oscar González-Cervantes
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  2. Juan Bory-Reyes
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  3. Irene Sabadini
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All authors contributed equally to the study, read and approved the final version of the submitted manuscript.

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Correspondence to Juan Bory-Reyes.

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González-Cervantes, J.O., Bory-Reyes, J. & Sabadini, I. Fractional Slice Regular Functions of a Quaternionic Variable. Results Math 79, 32 (2024). https://doi.org/10.1007/s00025-023-02047-6

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