Advertisement

On several unified reduction formulas for the Humbert function \(\Phi _{2}\) with applications

  • 31 Accesses

Abstract

In this paper, several unified reduction formulas for the Humbert function \(\Phi _{2}\) are given. They are further used to evaluate a new class of finite integrals involving Humbert function. Few known as well as new results have been obtained as special cases of our main findings.

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

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

References

  1. 1.

    Belafha, AAl, Nebdi, H.: Generation and propagation of novel donut beam by a spiral phase plate:Humbert Beams. Opt. Quant. Electron. 46, 201–208 (2014)

  2. 2.

    Brychkov, YuA: Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas. Chapman and Hall/CRC Press, Boca Raton (2008)

  3. 3.

    Brychokov, YuA: On some properties of Nu Hall function \(Q_v(a, b)\). Integral Transforms Spec. Funct. 25(1), 33–43 (2014)

  4. 4.

    Brychkov, YuA: Reduction formulas for the Appell and Humbert functions. Integral Transforms Spec. Funct. 28(1), 22–38 (2017)

  5. 5.

    Brychkov, Yu A., Kim, Yong Sup, Rathie, Arjun K.: On new reduction formulas for the Humbert functions \(\Psi _2\), \(\Phi _2\) and \(\Phi _3\). Integral Transforms Spec. Funct. 28(5), 350–360 (2017)

  6. 6.

    Brychkov, Yu., Saad, N.: On some formulas for the Appell Function \(F_1(a, b, b^{\prime }; c; \omega, z)\). Integral Transforms Spec. Funct. 23(11), 793–802 (2012)

  7. 7.

    Brychkov, Yu., Saad, N.: On some formulas for the Appell Function \(F_2(a, b, b^{\prime }; c, c^{\prime }; \omega, z)\). Integral Transforms Spec. Funct. 25(2), 111–123 (2014)

  8. 8.

    Brychkov, Yu., Saad, N.: On some formulas for the Appell Function \(F_3(a, a^{\prime }; b, b^{\prime }: c; \omega, z)\). Integral Transforms Spec. Funct. 26(11), 910–923 (2015)

  9. 9.

    Brychkv, YuA, Savischenko, N.V.: A special function of communication theorey. Integral Transforms Spec. Funct. 26(6), 470–484 (2015)

  10. 10.

    Brychkov, YuA, Savischenko, N.V.: Some properties of Owen T-function. Integral Transforms Spec. Funct. 27(2), 163–180 (2016)

  11. 11.

    Burchnall, J.L., Chaundy, T.W.: Expansions of Appell’s double hypergeometric functions. Quart. J. Math. (Oxford ser.) 11, 249–270 (1940)

  12. 12.

    Burchnall, J.L., Chaundy, T.W.: Expansions of Appell’s double hypergeometric functions II. Quart. J. Math. (Oxford ser.) 12, 112–128 (1941)

  13. 13.

    Bytev, V.V., Kalmykov, M.Yu., Moch, S.O.: HYPERDIRE: hypergeometric functions differential reduction mathematica based Packages for differential reduction of generalized hypergeometric functions : FD and FS Horn-type hypergeometric functions of three variables. Comput. Phys. Commun. 185, 3041–3058 (2014)

  14. 14.

    Choi, J., Rathie, A.K.: certain summation formulas for Humbert’s double hypergeometric series \(\Psi _2\) and \(\Phi _2\). Commun. Korean Math. Soc. 30(4), 439–446 (2015)

  15. 15.

    Krichi, B.A., Tarasov, O.V.: Finding new relationships between hypergeometric functions by evaluating Feynman integrals. Nucl. Phys. B 854(3), 841–852 (2012)

  16. 16.

    Makano, V.V.: A connection formula between double hypergeometric series \(\Psi _2\) and \(\Phi _3\). Integral Transforms Spec. Funct. 23(7), 503–508 (2012)

  17. 17.

    Prudnikov, A.P., Brychkov, YuA, Marichev, O.I.: Integrals and Series: More Special Functions, vol. 3. Gordon and Breach science Publ., New York (1960)

  18. 18.

    Rainville, E.D.: Special Functions. Macmillan Company, New York (1971)

  19. 19.

    Rathie, A.K.: On representation of Humbert’s double hypergeometric series in a series of Gauss’s \({}_2F_1\) function, arXiv:1312.0064v1, (2013)

  20. 20.

    Sofotasios, P.C., Tsiftsis, T.A., Brychkov, YuA, et al.: Analytic expressions and bounds for special functions and applications in communications theory. IEEE Trans. Inform. Theory 60(12), 7798–7823 (2014)

  21. 21.

    Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. Halsted Press, NewYork (1984)

  22. 22.

    Srivastava, H.M., Choi, J.: Zeta and q-zeta Functions and Associated series and integrals. Elsevier Science Publ, Amsterdam (2012)

Download references

Acknowledgements

All authors contributed equally in this paper. They have read and approved the final manuscript. The first author (YSK) acknowledges the support of the Wonkwang University Research Fund (2020). The authors are highly grateful to the referee for minutely reading the manuscript and pointing out certain typo corrections.

Author information

Correspondence to Medhat A. Rakha.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kim, Y., Rakha, M.A. & Rathie, A.K. On several unified reduction formulas for the Humbert function \(\Phi _{2}\) with applications. RACSAM 114, 62 (2020). https://doi.org/10.1007/s13398-020-00797-4

Download citation

Keywords

  • Humbert function
  • Appell function
  • Hypergeometric function
  • Reduction formulas

Mathematics Subject Classification

  • C33
  • 33C77
  • 33C70