Integral representations for a class of triple confluent hypergeometric functions and their applications in boundary value problems
DOI:
https://doi.org/10.31489/2026m1/140-155Keywords:
Gaussian hypergeometric function, Appell functions, Humbert functions, Srivastava–Karlsson hypergeometric functions, triple confluent hypergeometric series, integral representation, singular Helmholtz equation, application of hypergeometric functionsAbstract
Hypergeometric functions are divided into complete and confluent functions. Srivastava and Karlsson were the first to propose a method for constructing the complete set of triple Gaussian hypergeometric series and compiled a table containing definitions and regions of convergence for 205 distinct complete series in three variables. Subsequently, several authors obtained various integral representations and transformation formulas for the functions introduced by Srivastava and Karlsson. More recently, Ergashev identified 395 hypergeometric series of three variables that represent confluent forms of the known 205 complete hypergeometric series. In the present study, new Euler-type integral representations are derived for certain Gaussian hypergeometric functions of three variables. The main results are obtained using properties of the gamma and beta functions. New integral representations are established for 14 functions from the list of confluent hypergeometric functions of three variables. All derived integrals can be regarded as generalized Euler type representations of the classical Gaussian hypergeometric functions of one and two variables. In addition, it is demonstrated how one of these confluent functions, together with its integral representation, can be applied to construct solutions of the three-dimensional singular Helmholtz equation.
References
Brychkov, Y.A., & Savischenko, N.V. (2019). Application of hypergeometric functions of two variables in Wireless communication theory. Lobachevskii Journal of Mathematics, 40(7), 938– 953. https://doi.org/10.1134/S1995080219070096
Karlsson, P.W., & Ernst, T. (2024). Applications of quadratic and cubic hypergeometric transformations. Annales Universitatis Mariae Curie Sklodowska, Sectio A Mathematica, 78(1), 37–73. https://doi.org/10.17951/a.2024.78.1.37-73
Irgashev, B.Y. (2024). Application of hypergeometric functions to the construction of particular solutions. Complex Variables and Elliptic Equations, 69(12), 2025–2047. https://doi.org/10.1080/17476933.2023.2270910
Srivastava, H.M., & Karlsson, P.W. (1985). Multiple Gaussian Hypergeometric Series. Ellis Horwood Series in Mathematics and Its Applications. Chichester, West Sussex, England: E. Horwood; New York: Halsted Press.
Horn, J. (1931). Hypergeometrische Funktionen zweier Ver¨anderlichen. Mathematische Annalen, 105, 381–407. https://doi.org/10.1007/BF01455825
Humbert, P. (1922). The confluent hypergeometric functions of two variables. Proceedings of the Royal Society of Edinburgh, 41(1), 73–96.
Erdelyi, A., Magnus, W., Oberhettinger, F., & Tricomi, F.G. (Eds.). (1953). Higher Transcendental Functions, Volume I. New York, Toronto, London: McGraw-Hill Book Company, Inc.
Lauricella, G. (1893). Sulle funzione ipergeometriche a piu` variabili. Rendiconti del Circolo Matematico di Palermo, 7, 111–158. https://doi.org/10.1007/BF03012437
Saran, S. (1954). Hypergeometric functions of three variables. Ganita, 5, 77–91.
Sahai, V., & Verma, A. (2015). Recursion formulas for multivariable hypergeometric functions. Asian-European Journal of Mathematics, 8(4), Article 1550082. https://doi.org/10.1142/S1793557115500825
Sahai, V., & Verma, A. (2022). Infinite summation formulas of Srivastava’s general triple hypergeometric function. Mathematics in Engineering, Science and Aerospace (MESA), 13(4), 212–221.
Bezrodnykh, S.I. (2018). The Lauricella hypergeometric function , the Riemann–Hilbert problem, and some applications. Russian Mathematical Surveys, 73(6), 941–1031. https://doi.org/10.1070/RM9841
Bezrodnykh, S.I. (2018). Analytic continuation of the Lauricella function with arbitrary number of variables. Integral Transforms and Special Functions, 29(1), 21–42. https://doi.org/10.1080/10652469.2017.1402017
Bezrodnykh, S.I. (2022). Formulas for computing the Lauricella function in the case of crowding of variables. Computational Mathematics and Mathematical Physics, 62(12), 2069–2090. https://doi.org/10.1134/S0965542522120041
Srivastava, H.M. (1964). Hypergeometric functions of three variables. Ganita, 15, 97–108.
Srivastava, H.M. (1967). Some integrals representing triple hypergeometric functions. Rendiconti del Circolo Matematico di Palermo, 16, 99–115.
Choi, J., & Parmar, R.K. (2017). Generalized Srivastava’s triple hypergeometric functions and their associated properties. Journal of Nonlinear Sciences and Applications, 10(2), 817–827. https://doi.org/10.22436/jnsa.010.02.41
Ata, E. (2024). M-Srivastava hypergeometric functions: integral representations and solution of the fractional differential equations. Miskolc Mathematical Notes, 25(1), 93–108. https://doi.org/10.18514/MMN.2024.4326
Parmar, R.K., & Pog´any, T.K. (2017). Extended Srivastava’s triple hypergeometric HA,p,q function and related bounding inequalities. Journal of Contemporary Mathematical Analysis, 52(6), 276–287. https://doi.org/10.3103/S1068362317060036
Dar, S.A., & Paris, R.B. (2019). A (p,q)-extension of Srivastava’s triple hypergeometric function HB and its properties. Journal of Computational and Applied Mathematics, 348, 237–245. https://doi.org/10.1016/j.cam.2018.08.045
Dar, S.A., & Paris, R.B. (2020). A (p,q)-extension of Srivastava’s triple hypergeometric function HC. Publications De L Institut Mathematique, 108(122), 33–45. https://doi.org/10.2298/PIM 2022033D
Blu¨mlein, J., & Schneider, C. (2023). Hypergeometric structures in Feynman integrals. Annals of Mathematics and Artificial Intelligence, 91(5), 591–649. https://doi.org/10.1007/s10472-02309831-8
Hasanov, A., & Ruzhansky, M. (2019). Euler-type integral representations for the hypergeometric functions in three variables of second order. Bulletin of the Institute of Mathematics, 2(6), 73–223.
Bin-Saad, M.G., Shahwan, M.J.S., Younis, J.A., Aydi, H., & Abd El Salam, M.A. (2022). On Gaussian hypergeometric functions of three variables: Some new integral representations. Journal of Mathematics, 2022, Article 1914498. https://doi.org/10.1155/2022/1914498
Hasanov A., & Ruzhansky M. (2022). Systems of differential equations of Gaussian hypergeometric functions in three variables and their linearly-independent solutions. Bulletin of the Institute of Mathematics, 5(3), 50–142.
Jain, R.N. (1966). The confluent hypergeometric functions of three variables. Proceedings of the National Academy of Sciences, India, 36(2), 395–408.
Ergashev, T.G. (2024). Confluent hypergeometric functions in three variables and associated systems of partial differential equations. Bulletin of the Institute of Mathematics, 7(4), 121–215.
Srivastava, H.M., & Shpot, M.A. (2017). Reduction and transformation formulas for the Appell and related functions in two variables. Mathematical Methods in the Applied Sciences, 40(11), 4102–4108. https://doi.org/10.1002/mma.4289
Belafhal, A., Nossir, N., Dalil-Essakali, L., & Usman, T. (2020). Integral transforms involving the product of Humbert and Bessel functions and its application. AIMS Mathematics, 5(2), 1260–1274. https://doi.org/10.3934/math.2020086
Atash, A.A., & Bellahaj, H.S. (2018). Integrals formulas involving confluent hypergeometric functions of three variables Φ2 and Ψ2 . University of Aden Journal of Natural and Applied Sciences, 22(1), 143–149. https://doi.org/10.47372/uajnas.2018.n1.a11
Arzikulov, Z.O., Hasanov, A., & Ergashev, T.G. (2025). Konfliuentnye gipergeometricheskie funktsii i ikh primenenie k resheniiu zadachi Dirikhle dlia uravneniia Gelmgoltsa s tremia singuliarnymi koeffitsientami [Confluent hypergeometric functions and their application to the solution of Dirichlet problem for the Helmholtz equation with three singular coefficients.] Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seriia «Fiziko-matematicheskie nauki» — Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, 29(3), 407–429. https://doi.org/10.14498/vsgtu2156
Hasanov, A., & Yuldashova, H. (2025). Some functional identities of confluent hypergeometric function E1 of three variables. Bulletin of the Institute of Mathematics, 8(5), 18–29.
Kim, I., Paris, R.B., & Rathie, A.K. (2022). Some new results for the Kamp´e de F´eriet function with an application. Symmetry, 14(12), Article 2588. https://doi.org/10.3390/sym14122588
Choi, J., & Rathie, A.K. (2019). General summation formulas for the Kamp´e de F´eriet function. Montes Taurus Journal of Pure and Applied Mathematics, 1(1), 107–128.
Choi, J.J., Milovanovi´c, C.V., & Rathie, A.K. (2021). Generalized summation formulas for the Kamp´e de F´eriet functions. Axioms, 10(4), Article 318. https://doi.org/10.3390/axioms10040318
