Geodesic
Jacobi-Type Differential Relations for the Lauricella Function $F_D^{(N)}$
Matematičeskie zametki, Tome 99 (2016) no. 6, pp. 832-847.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: generalized Lauricella hypergeometric function, Jacobi-type differential relation, Jacobi identity, Gauss function, Christoffel–Schwarz integral. 
@article{MZM_2016_99_6_a2,
     author = {S. I. Bezrodnykh},
     title = {Jacobi-Type {Differential} {Relations} for the {Lauricella} {Function} $F_D^{(N)}$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {832--847},
     publisher = {mathdoc},
     volume = {99},
     number = {6},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2016_99_6_a2/}
}
TY  - JOUR
AU  - S. I. Bezrodnykh
TI  - Jacobi-Type Differential Relations for the Lauricella Function $F_D^{(N)}$
JO  - Matematičeskie zametki
PY  - 2016
SP  - 832
EP  - 847
VL  - 99
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2016_99_6_a2/
LA  - ru
ID  - MZM_2016_99_6_a2
ER  - 
%0 Journal Article
%A S. I. Bezrodnykh
%T Jacobi-Type Differential Relations for the Lauricella Function $F_D^{(N)}$
%J Matematičeskie zametki
%D 2016
%P 832-847
%V 99
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2016_99_6_a2/
%G ru
%F MZM_2016_99_6_a2
S. I. Bezrodnykh. Jacobi-Type Differential Relations for the Lauricella Function $F_D^{(N)}$. Matematičeskie zametki, Tome 99 (2016) no. 6, pp. 832-847. http://geodesic.mathdoc.fr/item/MZM_2016_99_6_a2/

[1] G. Lauricella, “Sulle funzioni ipergeometriche a piu variabili”, Rendiconti Circ. Math. Palermo, 7, Suppl. 1 (1893), 111–158 | DOI

[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 (1881), 305–322 | MR | Zbl

[3] P. Appel, J. Kampé de Feriet, Fonctions hypergéometriques et hypersphérique, Gauthier–Villars, Paris, 1926

[4] A. Erdélyi, “Hypergeometric functions of two variables”, Acta Math., 83 (1950), 131–164 | DOI | MR | Zbl

[5] O. M. Olson, “Integration of the partial differential equations for the hypergeometric function $F_1$ and $F_D$ of two and more variables”, J. Math. Phys., 5:3 (1964), 420–430 | DOI | MR

[6] H. Exton, Multiple Hypergeometric Functions and Application, John Wiley Sons, New York, 1976 | MR | Zbl

[7] P. Deligne, G. D. Mostow, “Monodromy of hypergeometric functions and nonlattice integral monodromy”, Publ. Math. Inst. Hautes Étud. Sci., 63 (1986), 5–89 | DOI | MR | Zbl

[8] U. Miller, Simmetriya i razdelenie peremennykh, Mir, M., 1981 | Zbl

[9] K. Iwasaki, H. Kimura, Sh. Shimomura, M. Yoshida, From Gauss to Painlevé. A Modern Theory of Special Functions, Aspects of Math., E16, Friedrich Vieweg Sohn, Braunschweig, 1991 | MR | Zbl

[10] M. E. H. Ismail, J. Pitman, “Algebraic evaluations of some Euler integrals, duplication formulae for Appell's hypergeometric function $F_1$, and Brownian variations”, Canad. J. Math., 52:5 (2000), 961–981 | MR | Zbl

[11] G. V. Kraniotis, General Relativity, Lauricella's Hypergeometric Function $F_D$ and the Theory of Braids, 2007, arXiv: 0709.3391

[12] R. R. Gontsov, “O podvizhnykh osobennostyakh sistem Garne”, Matem. zametki, 88:6 (2010), 845–858 | DOI | MR | Zbl

[13] T. A. Driscoll, L. N. Trefethen, Schwarz–Christoffel Mapping, Cambridge Monogr. Appl. Comput. Math., 8, Cambridge Univ. Press, Cambridge, 2002 | MR | Zbl

[14] S. I. Bezrodnykh, V. I. Vlasov, “Singulyarnaya zadacha Rimana–Gilberta v slozhnykh oblastyakh”, Spectral and Evolution Problems, 16 (2006), 112–118

[15] C. G. J. Jacobi, “Untersuchungen über die Differentialgleichungen der hypergeometrischen Reihe”, J. Reine Angew. Math., 56 (1859), 149–165 | DOI | MR | Zbl

[16] E. G. C. Poole, Introduction to the Theory of Linear Differential Equations, Clarendon Press, Oxford, 1936 | Zbl

[17] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii. Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Spravochnaya matematicheskaya biblioteka, Nauka, M., 1973 | MR | Zbl

[18] S. I. Bezrodnykh, “Sootnoshenie tipa Yakobi dlya obobschennoi gipergeometricheskoi funktsii”, III mezhdunarodnaya konferentsiya “Matematicheskie idei P. L. Chebysheva i ikh prilozhenie k sovremennym problemam estestvoznaniya” (Obninsk, 14–18 maya 2006 g.), Tezisy dokladov, 2006, 18–19

[19] S. I. Bezrodnykh, “Formuly analiticheskogo prodolzheniya i sootnosheniya tipa Yakobi dlya funktsii Laurichelly”, Dokl. RAN, 467:1 (2016), 7–12

[20] S. I. Bezrodnykh, Singulyarnaya zadacha Rimana–Gilberta i ee prilozhenie, Dis. $\dots$ kand. fiz.-matem. nauk, VTs RAN, M., 2006

[21] S. I. Bezrodnykh, V. I. Vlasov, “Singulyarnaya zadacha Rimana–Gilberta v slozhnykh oblastyakh”, Zh. vychisl. matem. i matem. fiz., 54:12 (2014), 1904–1953 | DOI | MR

[22] S. I. Bezrodnykh, V. I. Vlasov, B. V. Somov, “Obobschennye analiticheskie modeli tokovogo sloya Syrovatskogo”, Pisma v Astronom. zhurn., 37:2 (2011), 133–150

[23] S. I. Bezrodnykh, B. V. Somov, “An analysis of magnetic field and magnetosphere of neutron star under effect of a shock wave”, Adv. in Space Res., 56 (2015), 964–969 | DOI

[24] E. T. Uitteker, Dzh. N. Vatson, Kurs sovremennogo analiza. Ch. 2. Transtsendentnye funktsii, Editorial URSS, M., 2002 | Zbl

[25] P. Appel, “Sur les fonctions hypergéométriques de deux variables”, J. Math. Pure Appl. (3), 8 (1882), 173–216

[26] V. I. Vlasov, Kraevye zadachi v oblastyakh s krivolineinoi granitsei, Dis. $\dots$ dokt. fiz.-matem. nauk, VTs AN SSSR, M., 1990

[27] S. I. Bezrodnykh, “Ob analiticheskom prodolzhenii funktsii Laurichelly”, Mezhdunarodnaya konferentsiya po differentsialnym uravneniyam i dinamicheskim sistemam (Suzdal, 27 iyunya – 2 iyulya 2008 g.), Tezisy dokladov, 2008, 34–36

[28] Yu. A. Brychkov, N. Saad, “Some formulas for the Appell function $F_1 (a, b, b'; c; w, z)$”, Integral Transforms Spec. Funct., 23:11 (2012), 793–802 | DOI | MR | Zbl

[29] B. Riemann, “Beiträige zur Theorie der durch die Gauss'sche Reihe $F(\alpha, \beta, \gamma, x)$ darstellbaren Functionen”, Abh. Kön. Ges. d. Wiss. zu Göttingen, VII (1857) http://www.emis.de/classics/Riemann/PFunct.pdf