Geodesic
Expansion formulas for hypergeometric functions of two variables
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations, geometry, and topology, Tome 201 (2021), pp. 80-97.

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In the theory of hypergeometric functions, an important role is played by expansion formulas that allows one to express hypergeometric functions of several variables as infinite sums of products of several hypergeometric functions of one variable. In this paper, for hypergeometric functions of two variables, we introduce new symbolic Burchnall–Chaundy operators, examine their properties, and construct some expansions.
Keywords: hypergeometric function, expansion formula, symbolic form, Burchnall operator, Chaundy operator. 
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T. G. Ergashev. Expansion formulas for hypergeometric functions of two variables. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations, geometry, and topology, Tome 201 (2021), pp. 80-97. http://geodesic.mathdoc.fr/item/INTO_2021_201_a6/

[1] Beitmen A., Erdeii A., Vysshie transtsendentnye funktsii. T. 1, Nauka, M., 1973

[2] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integraly i ryady. Dopolnitelnye glavy, Nauka, M., 1986

[3] Urinov A. K., Ergashev T. G., “Konflyuentnye gipergeometricheskie funktsii mnogikh peremennykh i ikh primenenie k nakhozhdeniyu fundamentalnykh reshenii obobschennogo uravneniya Gelmgoltsa s singulyarnymi koeffitsientami”, Vestn. Tomsk. un-ta. Mat. Mekh., 55 (2018), 45–56

[4] Ergashev T. G., “Tretii potentsial dvoinogo sloya dlya obobschennogo dvuosesimmetricheskogo uravneniya Gelmgoltsa”, Ufim. mat. zh., 10:4 (2018), 111–122 | MR | Zbl

[5] Ergashev T. G., “Chetvertyi potentsial dvoinogo sloya dlya obobschennogo dvuosesimmetricheskogo uravneniya Gelmgoltsa”, Vestn. Tomsk. un-ta. Mat. Mekh., 50 (2017), 45–56

[6] Appell P. and Kampe de Feriet J., Fonctions Hypergeometriques et Hyperspheriques. Polynomes d'Hermite, Gauthier-Villars, Paris, 1926

[7] Berdyshev A. S, Hasanov A., Ergashev T. G., “Double-layer potentials for a generalized biaxially symmetric Helmholtz equation, II”, Compl. Var. Elliptic Equations., 65 (2019), 1–17

[8] Burchnall J. L., Chaundy T. W., “Expansions of Appell's double hypergeometric functions”, Q. J. Math. Oxford., 11 (1940), 249–270 | DOI | MR

[9] Burchnall J. L., Chaundy T. W., “Expansions of Appell's double hypergeometric functions, II”, Q. J. Math. Oxford., 12 (1941), 112–128 | DOI | MR

[10] Erdelyi A., Magnus W., Oberhettinger F., Tricomi F. G., Higher Transcendental Functions. Vol. I, McGraw-Hill, New York–Toronto–London, 1953 | Zbl

[11] Ergashev T. G., “On fundamental solutions for multidimensional Helmholtz equation with three singular coefficients”, Comput. Math. Appl., 77 (2019), 69–76 | MR | Zbl

[12] Ergashev T. G., Hasanov A., “Fundamental solutions of the bi-axially symmetric Helmholtz equation”, Uzbek Math. J., 1 (2018), 55–64 | DOI | Zbl

[13] Exton H., “On certain confluent hypergeometric functions of three variables”, Ganita, 21:2 (1970), 79–92 | MR | Zbl

[14] Hasanov A., “Fundamental solutions bi-axially symmetric Helmholtz equation”, Compl. Var. Elliptic Equations., 52:8 (2007), 673–683 | DOI | Zbl

[15] Horn J., “Über die Convergenz der hypergeometrischen Reihen zweier und dreier Veränderlichen”, Math. Ann., 34 (1889), 544–600 | DOI | MR

[16] Horn J., “Hypergeometrische Funktionen zweier Veränderlichen”, Math. Ann., 105 (1931), 381–407 | DOI | MR

[17] Jain R. N., “The confluent hypergeometric functions of three variables”, Proc. Natl. Acad. Sci. India Sect. A., 36 (1966), 395–408 | Zbl

[18] Karimov E. T., Nieto J. J., “The Dirichlet problem for a 3D elliptic equation with two singular coefficients”, Comput. Math. Appl., 62 (2011), 214–224 | DOI | MR | Zbl

[19] Mavlyaviev R. M., Garipov I. B., “Fundamental solution of multidimensional axisymmetric Helmholtz equation”, Compl. Var. Elliptic Equations., 62:3 (2016), 287–296 | DOI | MR

[20] Salakhitdinov M. S., Karimov E. T., “Spatial boundary problem with the Dirichlet-Neumann condition for a singular elliptic equation”, Appl. Math. Comput., 219 (2012), 3469–3476 | MR | Zbl

[21] Srivastava H. M., Hasanov A., Choi J., “Double-layer potentials for a generalized bi-axially symmetric Helmholtz equation”, Sohag J. Math., 2:1 (2015), 1–10

[22] Srivastava H. M., Karlsson P. W., Multiple Gaussian Hypergeometric Series, Halsted Press, New York, 1985 | Zbl

[23] Srivastava H. M., Manocha H. L., A Treatise on Generating Functions, Halsted Press, New York, 1984 | Zbl