Mathematical Notes

, Volume 99, Issue 5–6, pp 821–833 | Cite as

Jacobi-type differential relations for the Lauricella function F D (N)

  • S. I. Bezrodnykh
Short Communications


For the generalized Lauricella hypergeometric function F D (N) , Jacobi-type differential relations are obtained and their proof is given. A new system of partial differential equations for the function F D (N) is derived. Relations between associated Lauricella functions are presented. These results possess a wide range of applications, including the theory of Riemann–Hilbert boundary-value problem.


generalized Lauricella hypergeometric function Jacobi-type differential relation Jacobi identity Gauss function Christoffel–Schwarz integral 


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  1. 1.
    G. Lauricella, “Sulle funzioni ipergeometriche a piu variabili,” Rendiconti Circ. Math. Palermo 7, 111–158 (1893), Suppl. 1.CrossRefGoogle Scholar
  2. 2.
    E. Picard, “Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques,” Ann. Sci. école Norm. Sup. (2) 10, 305–322 (1881).Google Scholar
  3. 3.
    P. Appel and J. Kampéde Feriet, Fonctions hypergéometriques et hypersphérique (Gauthier–Villars, Paris, 1926).Google Scholar
  4. 4.
    A. Erdélyi, “Hypergeometric functions of two variables,” ActaMath. 83, 131–164 (1950).MathSciNetzbMATHGoogle Scholar
  5. 5.
    O. M. Olson, “Integration of the partial differential equations for the hypergeometric function F1 and FD of two and more variables,” J.Math. Phys. 5 (3), 420–430 (1964).CrossRefGoogle Scholar
  6. 6.
    H. Exton, Multiple Hypergeometric Functions and Application (JohnWiley & Sons, New York, 1976).zbMATHGoogle Scholar
  7. 7.
    P. Deligne and G. D. Mostow, “Monodromy of hypergeometric functions and nonlattice integral monodromy,” Publ.Math. Inst. Hautes étud. Sci. 63, 5–89 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    W. Miller, Symmetry Separation of Variables (Addison–Wesley, Reading (USA), 1977; Mir, Moscow, 1981).Google Scholar
  9. 9.
    K. Iwasaki, H. Kimura, Sh. Shimomura, and M. Yoshida, From Gauss to Painlevé. A Modern Theory of Special Functions, in Aspects of Math. (Friedrich Vieweg & Sohn, Braunschweig, 1991), Vol. E16.zbMATHGoogle Scholar
  10. 10.
    M. E. H. Ismail and J. Pitman, “Algebraic evaluations of some Euler integrals, duplication formulae for Appell’s hypergeometric function F1, and Brownian variations,” Canad. J. Math. 52 (5), 961–981 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    G. V. Kraniotis, General Relativity, Lauricella’s Hypergeometric Function FD, and the Theory of Braids, arXiv: 0709.3391 (2007).Google Scholar
  12. 12.
    R. R. Gontsov, “On movable singularities of Garnier systems,” Mat. Zametki 88 (6), 845–858 (2010) [Math. Notes 88 (5–6), 806–818 (2010)].MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    T. A. Driscoll and L. N. Trefethen, Schwarz–Christoffel Mapping, in CambridgeMonogr. Appl. Comput. Math. (Cambridge Univ. Press, Cambridge, 2002), vol. 8.Google Scholar
  14. 14.
    S. I. Bezrodnykh and V. I. Vlasov, “The singular Riemann–Hilbert problem in the complicated domains,” Spectral and Evolution Problems 16, 112–118 (2006).Google Scholar
  15. 15.
    C. G. J. Jacobi, “Untersuchungen Über die Differentialgleichungen der hypergeometrischen Reihe,” J. Reine Angew. Math. 56, 149–165 (1859).MathSciNetCrossRefGoogle Scholar
  16. 16.
    E. G. C. Poole, Introduction to the Theory of Linear Differential Equations (Clarendon Press, Oxford, 1936).zbMATHGoogle Scholar
  17. 17.
    H. Bateman and A. Erdélyi, Higher Transcendental Functions, vol. 1: The Hypergeometric Function, Legendre Functions (McGraw–Hill, New York–Toronto–London, 1953; Nauka, Moscow, 1973 (2nd ed.)).zbMATHGoogle Scholar
  18. 18.
    S. I. Bezrodnykh, “A Jacobi-type relation for the generalized hypergeometric function,” in III International Conference “Mathematical Ideas of P. L. Chebyshev and Their Application to Modern Problems of Natural Science”, Obninsk, May 14–18, 2006, Abstracts of papers (2006), pp. 18–19 [in Russian].Google Scholar
  19. 19.
    S. I. Bezrodnykh, “Analytic continuation formulas and Jacobi-type relations for the Lauricella function,” Dokl. Ross. Akad. Nauk 467 (1), 7–12 (2016) [Dokl.Math 93 (2), 129–134 (2016)].Google Scholar
  20. 20.
    S. I. Bezrodnykh, Singular Riemann–Hilbert Problem and Its Application, Cand. Sci. (Phys.–Math.) Dissertation (Computer Center, Russian Academy of Sciences, Moscow, 2006) [in Russian].Google Scholar
  21. 21.
    S. I. Bezrodnykh and V. I. Vlasov, “Singular Riemann-Hilbert problem in complex-shaped domains,” Zh. Vychisl. Mat. i Mat. Fiz. 54 (12), 1904–1953 (2014) [Comput. Math. Math. Phys. 54 (12), 1826–1875 (2014)].MathSciNetzbMATHGoogle Scholar
  22. 22.
    S. I. Bezrodnykh, V. I. Vlasov„ and B. V. Somov, “Generalized Analytic Models of the Syrovatskii Current Layer,” Astronomy Letters 37 (2), 113–130 (2011).CrossRefGoogle Scholar
  23. 23.
    S. I. Bezrodnykh and B. V. Somov, “An analysis of the magnetic field and the magnetosphere of a neutron star under the effect of a shock wave,” Adv. in Space Res. 56, 964–969 (2015).CrossRefGoogle Scholar
  24. 24.
    E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Pt. 2: Transcendental Functions (Cambridge Univ. Press, Cambridge, 1996; Editorial URSS, Moscow, 2002).Google Scholar
  25. 25.
    P. Appel, “Sur les fonctions hypergéométriques de deux variables,” J. Math. Pure Appl. (3) 8, 173–216 (1882).Google Scholar
  26. 26.
    V. I. Vlasov, Boundary-Value Problems inDomainswith Curvilinear Boundary, Doctoral (Phys.–Math.) Dissertation (Computer Center, AN SSSR, Moscow, 1990) [in Russian].Google Scholar
  27. 27.
    S. I. Bezrodnykh, “On the analytic continuation of the Lauricella function,” in International Conference on Differential Equations and Dynamical Systems, Suzdal, June 27–July 2, 2008, Abstracts of papers (2008), pp. 34–36 [in Russian].Google Scholar
  28. 28.
    Yu. A. Brychkov and N. Saad, “Some formulas for the Appell function F1(a, b, b; c; w, z),” Integral Transforms Spec. Funct. 23 (11), 793–802 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    B. Riemann, “Beiträge zur Theorie der durch die Gaußs’sche Reihe F(a, ß, x) darstellbaren Functionen,” Abh. Kön. Ges. d. Wiss. zu Göttingen VII (1857), http:// wwwemisde/classics/Riemann/ PFunctpdf.Google Scholar

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© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Federal Research Center “Computer Science and Control,”Russian Academy of SciencesMoscowRussia
  2. 2.Sternberg State Astronomical InstituteLomonosov Moscow State UniversityMoscowRussia
  3. 3.Peoples’ Friendship University of RussiaMoscowRussia

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