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Generalized Krätzel functions: an analytic study

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Abstract

The paper is devoted to the study of generalized Krätzel functions, which are the kernel functions of type-1 and type-2 pathway transforms. Various analytical properties such as Lipschitz continuity, fixed point property and integrability of these functions are investigated. Furthermore, the paper introduces two new inequalities associated with generalized Krätzel functions. The composition formulae for various fractional operators, such as Grünwald-Letnikov fractional differential operator, Riemann-Liouville fractional integral and differential operators with these functions are also obtained. Additionally, it has been demonstrated that these kernel functions are related to the Heaviside step function, the Dirac delta function, and the Riemann zeta function. A computable series representation of the kernel function is also obtained. Further scope of the work is pointed out by modifying the kernel function with the Mittag-Leffler function.

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References

  1. Arfken, G.B., Weber, H.J.: Mathematical Methods for Physicists, 5th edn. Academy Press, Massachusetts (2000)

    Google Scholar 

  2. Baricz, Á., Jankov, D., Pogány, T.K.: Turán type inequality for Krätzel functions. J. Math. Anal. Appl. 388, 716–724 (2012). https://doi.org/10.1016/j.jmaa.2011.09.057

    Article  MathSciNet  Google Scholar 

  3. Bonilla, B., Rivero, M., Rodrigues, J., Trujillo, J.J., Kilbas, A.A.: Bessel-type functions and Bessel-type integral transforms on spaces \(\cal{F} _{p,\mu }\) and \(\cal{F} _{p,\mu }^{^{\prime }}\). Integral Transform Spec. Funct. 8(1–2), 13–30 (1999). https://doi.org/10.1080/10652469908819213

    Article  MathSciNet  Google Scholar 

  4. Conway, J.B.: Functions of One Complex Variable I. \(2^{nd}\) Ed. Springer. New York (1978)

  5. Dimovski, I., Kiryakova, V.: The Obrechkoff integral transform: Properties and relation to a generalized fractional calculus. Numer. Funct. Anal. and Optimiz. 21(1–2), 121–144 (2000). https://doi.org/10.1080/01630560008816944

    Article  MathSciNet  Google Scholar 

  6. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions. Vol. II, New York, Toronto, London, McGraw-Hill (1954) (Reprinted: Krieger, Melbourne, Florida (1981)

  7. Glaeske, H.J., Kilbas, A.A., Saigo, M.: A modified Bessel-type integral transform and its compositions with fractional calculus operators on spaces \(\cal{F} _{p,\mu }\) and \(\cal{F} ^{^{\prime }}_{p,\mu }\). J. Comput. Appl. Math. 118, 151–168 (2000). https://doi.org/10.1016/S0377-0427(00)00286-7

    Article  ADS  MathSciNet  Google Scholar 

  8. Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin S.: Mittag-Leffler Functions, Related Topics and Applications. \(2^{nd}\) Ed. Springer Monographs in Mathematics (2020)

  9. Haubold, H.J., Kumar, D.: Extension of thermonuclear functions through the pathway model including Maxwell-Boltzmann and Tsallis’ distributions. Astropart. Physics 29, 70–76 (2008). https://doi.org/10.1016/j.astropartphys.2007.11.006

    Article  ADS  Google Scholar 

  10. Haubold, H.J., Kabeer, A.A., Kumar, D.: Analytic forms of thermonuclear functions. Physica A 630, 129249 (2023). https://doi.org/10.1016/j.physa.2023.129249

    Article  MathSciNet  Google Scholar 

  11. Kilbas, A.A., Kumar, D.: On generalized Krätzel function. Integral Transforms Spec. Funct. 20, 835–846 (2009). https://doi.org/10.1080/10652460902819024

    Article  MathSciNet  Google Scholar 

  12. Kilbas, A.A., Saxena, R.K., Trujillo, J.J.: Krätzel function as a function of hypergeometric type. Fract. Calc. Appl. Anal. 9(2), 109–132 (2006)

    MathSciNet  Google Scholar 

  13. Kilbas, A.A., Rodríguez-Germá, L., Saigo, M., Saxena, R.K., Trujillo, J.J.: The Krätzel function and evaluation of integrals. Comput. Math. Appl. 59(5), 1790–1800 (2010). https://doi.org/10.1016/j.camwa.2009.08.017

    Article  MathSciNet  Google Scholar 

  14. Kiryakova, V.: Generalized Fractional Calculus and Applications. Longman & J. Wiley, Harlow, New York (1994)

    Google Scholar 

  15. Kumar, D., Haubold, H.J.: On extended thermonuclear functions through pathway model. Adv. Space Res. 45, 698–708 (2009). https://doi.org/10.1016/j.asr.2009.07.033

    Article  ADS  CAS  Google Scholar 

  16. Kumar, D., Haubold, H.J.: Analytic representation of Maxwell-Boltzmann and Tsallis’ thermonuclear functions with depleted tail. Axioms 10(115), (2021). https://doi.org/10.3390/axioms10020115

  17. Kumar, D., Kilbas, A.A.: Fractional Calculus of \(\cal{P} \)-Transforms. Fract. Calc. Appl. Anal. 13, 309–328 (2010)

    MathSciNet  Google Scholar 

  18. Kumar, D.: \(\cal{P} \)-transform. Integral Transforms Spec. Funct. 22, 603–616 (2011). https://doi.org/10.1080/10652469.2010.536.410

    Article  ADS  MathSciNet  Google Scholar 

  19. Krätzel, E.: Integral transformations of Bessel type, in Generalized Functions and Operational Calculus. (Proc. Conf. Varna, 1975), Bulgarian Academy of Science, Sofia, 148-155 (1979)

  20. Mathai, A.M.: A Pathway to matrix-variate gamma and normal densities. Linear Algebra Appl. 396, 317–328 (2005). https://doi.org/10.1016/j.laa.2004.09.022

    Article  ADS  MathSciNet  Google Scholar 

  21. Mathai, A.M., Haubold, H.J.: Mathematical aspects of Krätzel and Krätzel transform. Mathematics 8, 526 (2020). https://doi.org/10.3390/math8040526

    Article  Google Scholar 

  22. Mathai, A.M., Haubold, H.J.: Pathway model, superstatistics, Tsallis’ statistics and a generalized measure of entropy. Physica A 375, 110–122 (2007). https://doi.org/10.1016/j.physa.2006.09.002

    Article  ADS  MathSciNet  Google Scholar 

  23. Mitrinović, D.S.: Analytic Inequalities. Springer-Verlag, Berlin (1970)

    Book  Google Scholar 

  24. Mihai, M.V., Awan, M.U., Noor, M.A., Noor, K.I., Kim, J.K.: Hermite-Hadamard inequalities and their applications. J. Inequal. Appl. 309, 1–9 (2018). https://doi.org/10.1186/s13660-018-1895-4

    Article  MathSciNet  Google Scholar 

  25. Nair, D.H., Kumar, D.: Treatise on generalized Krätzel function. Vietnam J. Math. 43, 23–36 (2015). https://doi.org/10.1007/s10013-014-0058-2

    Article  MathSciNet  Google Scholar 

  26. Ortigueira, M.D.: Fractional differences integral representation and its use to define fractional derivatives. Proceedings of the Fifth EUROMECH Nonlinear Dynamics Conference (ENOC-2005). Eindhoven. 1333-1337 (2005)

  27. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, New York (1993)

    Google Scholar 

  28. Schiff, J.L.: The Laplace Transform. Theory and Applications. Springer, New York (1999)

    Book  Google Scholar 

  29. Tassaddiq, A., Srivastava, R.: New results involving the generalized Krätzel function with application to the fractional kinetic equations. Mathematics 11, 1060 (2023). https://doi.org/10.3390/math11041060

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank the University of Kerala for providing all facilities, and the second author would like to thank the University of Kerala for the financial support under the start-up grand No. 1622/2021/UOK. The authors would like to acknowledge the unknown referees for their valuable comments and suggestions, which enhanced the paper to its present form.

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  1. Department of Mathematics, University of Kerala, Kariavattom, Thiruvananthapuram, Kerala, 695581, India

    Ashik A. Kabeer & Dilip Kumar

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  1. Ashik A. Kabeer
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  2. Dilip Kumar
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Correspondence to Dilip Kumar.

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Kabeer, A.A., Kumar, D. Generalized Krätzel functions: an analytic study. Fract Calc Appl Anal 27, 799–817 (2024). https://doi.org/10.1007/s13540-024-00243-x

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