Skip to main content
SpringerLink
Log in

A family of hyper-Bessel functions and convergent series in them

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

The Delerue hyper-Bessel functions that appeared as a multi-index generalizations of the Bessel function of the first type, are closely related to the hyper-Bessel differential operators of arbitrary order, introduced by Dimovski. In this work we consider an enumerable family of hyper-Bessel functions and study the convergence of series in such a kind of functions. The obtained results are analogues to the ones in the classical theory of the widely used power series, like Cauchy-Hadamard, Abel and Fatou theorem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Delerue, Sur le calcul symbolic à n variables et fonctions hyperbess éliennes (II). Annales Soc. Sci. Bruxelle Ser. 1, No 3 (1953), 229–274.

    Google Scholar 

  2. I. Dimovski, Operational calculus for a class of differential operators. Compt. Rend. Acad. Bulg. Sci. 19, No 12 (1966), 1111–1114.

    MathSciNet  Google Scholar 

  3. I. Dimovski, V. Kiryakova, Generalized Poisson transmutations and corresponding representations of hyper-Bessel functions. Compt. Rend. Acad. Bulg. Sci. 39, No 10 (1986), 29–32.

    MATH  MathSciNet  Google Scholar 

  4. I. Dimovski, V. Kiryakova, Generalized Poisson representations of hypergeometric functions p F q, p < q, using fractional integrals. In: Proc. 16th Spring Conf. Union Bulg. Math., Sofia (1987), 205–212.

    Google Scholar 

  5. A.A. Kilbas, A.A. Koroleva, S.V. Rogosin, Multi-parametric Mittag-Leffler functions and their extension. Fract. Calc. Appl. Anal. 16, No 2 (2013), 378–404; DOI:10.2478/s13540-013-0024-9; http://link.springer.com/article/10.2478/s13540-013-0024-9.

    Article  MathSciNet  Google Scholar 

  6. V. Kiryakova, Generalized Fractional Calculus and Applications. Longman & J. Wiley, Harlow-N. York (1994).

    MATH  Google Scholar 

  7. V. Kiryakova, All the special functions are fractional differintegrals of elementary functions. J. Physics A: Math. & General 30, No 14 (1997), 5085–5103; doi: 10.1088/0305-4470/30/14/019.

    Article  MATH  MathSciNet  Google Scholar 

  8. V. Kiryakova, Multiindex Mittag-Leffler functions, related Gelfond-Leontiev operators and Laplace type integral transforms. Fract. Calc. Appl. Anal. 2, No 4 (1999), 445–462.

    MATH  MathSciNet  Google Scholar 

  9. V. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. J. Comput. and Applied Mathematics 118 (2000), 241–259; doi:10.1016/S0377-0427(00)00292-2.

    Article  MATH  MathSciNet  Google Scholar 

  10. V. Kiryakova, The multi-index Mittag-Leffler functions as important class of special functions of fractional calculus. Computers and Mathematics with Appl. 59, No 5 (2010), 1885–1895; doi:10.1016/j.camwa.2009.08.025.

    Article  MATH  MathSciNet  Google Scholar 

  11. V. Kiryakova, V. Hernandez-Suarez, Bessel-Clifford third order differential operator and corresponding Laplace type integral transform. Dissertationes Mathematicae 340 (1995), 143–161.

    MATH  MathSciNet  Google Scholar 

  12. M. Kljuchantzev, On the construction of r-even solutions of singular differential equations. Dokladi AN SSR 224, No 5 (1975), 1000–1008 (In Russian).

    Google Scholar 

  13. M. Kljuchantzev, An introduction to the theory of (ν 1, …, ν r−1)-transforms. Mat. Sbornik 132, No 2 (1987), 167–181 (In Russian).

    Google Scholar 

  14. Yu. Luchko, Operational method in fractional calculus. Fract. Calc. Appl. Anal. 2, No 4 (1999), 463–488.

    MATH  MathSciNet  Google Scholar 

  15. O. I. Marichev, Method of Calculation of Integrals of Special Functions (in Russian). Nauka i Technika, Minsk (1978).

    Google Scholar 

  16. A. Markushevich, A Theory of Analytic Functions, 1, 2 (In Russian). Nauka, Moscow (1967).

  17. J. Paneva-Konovska, The convergence of series in multi-index Mittag-Leffler functions. Integral Transforms Spec. Funct. 23, No 3 (2012), 207–221; DOI:10.1080/10652469.2011.575567.

    Article  MATH  MathSciNet  Google Scholar 

  18. J. Paneva-Konovska, Fatou type theorems for series in Mittag-Leffler functions. In: AIP Conf. Proc. 1497, New York (2012), 318–325; doi: 10.1063/1.4766800.

    Article  Google Scholar 

  19. J. Paneva-Konovska, Comparison between the convergence of power and Bessel series. In: AIP Conf. Proc. 1570, New York (2013), 383–392; doi: 10.1063/1.4854780.

    Article  Google Scholar 

  20. J. Paneva-Konovska, On the multi-index (3m-parametric) Mittag-Leffler functions, fractional calculus relations and series convergence. Central European J. of Physics 11, No 10 (2013), 1164–1177; DOI: 10.2478/s11534-013-0263-8.

    Article  Google Scholar 

  21. P. Rusev, Classical Orthogonal Polynomials and Their Associated Functions in Complex Domain. Publ. House Bulg. Acad. Sci., Sofia (2005).

    MATH  Google Scholar 

  22. S. Yakubovich, Yu. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions. Kluwer Acad. Publ., Dordrecht — Boston — London (1994).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Applied Mathematics and Informatics, Technical University of Sofia, 8 ”Kliment Ohridski” bul., 1000, Sofia, Bulgaria

    Jordanka Paneva-Konovska

  2. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, ”Acad. G. Bontchev” Street, Block 8, Sofia, 1113, Bulgaria

    Jordanka Paneva-Konovska

Authors
  1. Jordanka Paneva-Konovska
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jordanka Paneva-Konovska.

Additional information

Dedicated to Professor Ivan Dimovski on the occasion of his 80th anniversary

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Paneva-Konovska, J. A family of hyper-Bessel functions and convergent series in them. Fract Calc Appl Anal 17, 1001–1015 (2014). https://doi.org/10.2478/s13540-014-0211-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s13540-014-0211-3

MSC 2010

Key Words and Phrases

Search

Navigation