IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v258y2015icp454-464.html
   My bibliography  Save this article

Differentiation formulas of some hypergeometric functions with respect to all parameters

Author

Listed:
  • Kang, Hongchao
  • An, Congpei

Abstract

In this work we present two methods to derive some differentiation formulas of the generalized hypergeometric function mFn(a1,…,am;b1,…,bn;z), including the most commonly used Gauss hypergeometric function 2F1(μ,ν;λ;z) and Kummer confluent hypergeometric function 1F1(μ;ν;z) as special cases, with respect to all parameters. We first briefly describe the direct derivative method for the convergent power series of hypergeometric functions. Secondly, we mainly focus on the differential equation method, which is based on differentiating the generalized hypergeometric differential equation with respect to parameters. Particularly, by using the differential equation method, some general analytical expressions of any sth derivatives with respect to single parameter can be deduced by induction in s. Moreover, we can obtain all the mixed derivatives of higher order very conveniently. Finally, examples are given to illustrate the usefulness of these derivatives in mathematics, physics and other related fields. Numerical examples for computing those singular oscillatory integrals presented in Kang et al. (2013) and Kang and Ling (in press), in turn verify that the approximation value of the required derivatives can be of great precision, and show the correctness of differentiation formulas obtained by the proposed methods.

Suggested Citation

  • Kang, Hongchao & An, Congpei, 2015. "Differentiation formulas of some hypergeometric functions with respect to all parameters," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 454-464.
    • Handle: RePEc:eee:apmaco:v:258:y:2015:i:c:p:454-464
    DOI: 10.1016/j.amc.2015.02.017
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300315001708
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2015.02.017?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Kang, Hongchao & Xu, Qi, 2023. "Quadrature formulae of many highly oscillatory Fourier-type integrals with algebraic or logarithmic singularities and their error analysis," Applied Mathematics and Computation, Elsevier, vol. 442(C).
    2. Kang, Hongchao, 2019. "Efficient calculation and asymptotic expansions of many different oscillatory infinite integrals," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 305-318.
    3. Li, Bin & Kang, Hongchao & Chen, Songliang & Ren, Shanjing, 2023. "On the approximation of highly oscillatory Volterra integral equations of the first kind via Laplace transform," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 214(C), pages 92-113.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:258:y:2015:i:c:p:454-464. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.