TARU PUBLICATIONS
Journal of Interdisciplinary Mathematics cover
Open Access ·Peer-reviewed·ISSN (Online): 2169-012X·ISSN (Print): 0972-0502

Monthly Journal: Publishes the methodological and theoretical role of mathematics and mathematical applications underpinning scientific research.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
submissions@tarupublications.com
Open Access Research Article

Certain properties of the Wright function

, *

* Corresponding author · click or hover a name for details

pp. 909–925Vol. 29Issue 4April 2026DOI: 10.47974/JIM-2380XML
Received:
01 Feb 2025
Published Online:
07 Apr 2026
Article type:
Research Article
Language:
EN
Article no.:
JIM-2380
Pages:
909–925

Abstract

In this paper, we have demonstrated several results for the Wright functions. We studied some basic properties, successive differentiation recurrence relations, various types of integral representations, integral duplication formulae, multiplication theorem, etc. Several particular cases give important results in the forms of well known Special Functions. 

Keywords

Subject Classifications

33E1233E2033E9933C99

References

[1] E. M. Wright, “On the Coefficients of Power Series Having Exponential Singularities”, J. London Math. Soc., s1-8, pp.71-79 (1933).
[2] E. M. Wright, “The Asymptotic Expansion of the Generalized Bessel Function”, Proc. Lond. Math. Soc., s2-38: pp.257-270 (1935), https://doi.org/10.1112/plms/s2-38.1.257.
[3] E. M. Wright, “The generalized Bessel function of order greater than one”, Q. J. Math.,  vol. 11, Issue 1, pp.36-48 (1940), https://doi.org/10.1093/qmath/os-11.1.36.
[4] E. M. Wright, “The Asymptotic Expansion of the Generalized Hypergeometric Function”, J. London Math. Soc. (2), s1-10, pp.286-293 (1935). 
[5] K. S. Gehlot, J. C. Prajapati, and B. P. Patel, “Differential recurrence relation of Generalized K- Wright function”, Int. J. Pure Appl. Math., vol. 87, no. 4, pp. 611-619 (2013), http://dx.doi.org/10.12732/ijpam.v87i4.10. 
[6] Y. Luchko, “The Wright function and its applications”, vol. 1, Basic Theory edited by Anatoly Kochubei and Yuri Luchko, pp. 241-268.Berlin, Boston: De Gruyter (2019), https://doi.org/10.1515/9783110571622-010.
[7] K. Mehrez, “New Integral representations for the Fox-Wright functions and its applications”, J. Math. Anal. Appl., vol. 468, no. 2, pp.650-673 (2018), https://doi.org/10.1016/j.jmaa.2018.08.053.
[8] K. Mehrez, and S. M. Sitnik, “Functional inequalities for Fox-Wright functions”, Ramanujan J, vol. 50, pp. 263-287 (2019), https://doi.org/10.1007/s11139-018-0071-2.
[9] F. Mainardi, and A. Consiglio, “The Wright Functions of the Second Kind in Mathematical Physics”, Mathematics, vol. 8, no. 6, 2020, https://doi.org/10.3390/math8060884.
[10] R. Gorenflo, Y. Luchko, and F. Mainardi, “Wright functions as scale-invariant solutions of the diffusion-wave equation”, J. Comput. Appl. Math., vol. 118, no. 1, pp.175-191 (2000), https://doi.org/10.1016/S0377-0427(00)00288-0.
[11] Y. Singh, and R. S. Dubey, “Fractional calculus operator with generalize k-Mittag-Leffler function”, J. Interdiscip. Math., vol. 23, no. 2, pp. 545-553 (2020), https://doi.org/10.1080/09720502.2020.1731971.
[12] R. S. Dubey, Y. Singh, P. Agarwal, G. L. Saini, and V. Singh, “Some results on the new fractional derivative of generalized k-Wright function” J. Interdiscip. Math., vol. 23, no. 2, pp. 607-615 (2020), https://doi.org/10.1080/09720502.2020.1731981.
[13] Liya D’Costa, and N. Menaria, “Fractional calculus of extended k-generalized Mittag-Leffler function and its properties”, J. Interdiscip. Math., vol. 27, no. 8, pp. 1741-1748 (2024), https://doi.org/10.47974/JIM-2036.
[14] E. D. Rainville , Special functions, The Macmillan Company, (1960). 
[15] A. Wiman, “Über den Fundamentalsatz in der Teorie der Funktionen Eα (x)”, Acta Math., vol. 29, pp.191-201 (1905), https://doi.org/10.1007/BF02403202.
[16] R. Gorenflo, A. Kilbas, F. Mainardi, and S. Rogosin, “The Two-Parametric Mittag-Leffler Function”, Mittag-Leffler Functions, Related Topics and Applications; Springer: Berlin, Germany, 2nd Edition, pp.63-113 (2020), https://doi.org/10.1007/978-3-662-61550-8_4. 
[17] G. N. Watson, “A Treatise on the Theory of Bessel Functions”, Amer. Math. Monthly, vol. 30, no. 6, pp. 326-327 (1923), https://doi.org/10.2307/2300274.
[18] P. Van Mieghem, “The Mittag-Leffler function”, arXiv e-prints, pp. 105-107 (2020), arXiv:2005.13330[math.FA], https://ui.adsabs.harvard.edu/abs/2020arXiv200513330V.
[19] A. Apelblat and J. L. González-Santand, “The integral Mittag-Leffler, Whittaker and Wright functions”, Mathematics, vol. 9, no. 24 (2021), https://doi.org/10.3390/math9243255.
[20] L. Andrews, “Other Functions Defined by Integrals”, Special Functions of Mathematics for Engineers, Second Edition, pp. 109-140 (1997), https://doi.org/10.1117/3.270709.ch3. 
[21] J. Riordan, “Combinatorial Identities”, Wiley Series in Probability and Mathematical Statistics, Wiley, (1968). 

Views: 144Downloads: 32Citations: 0