Geodesic
Quadratic transformations of multiple hypergeometric series
Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 2, pp. 517-531.

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Using factorization method and introducing canonical forms of multiple hypergeometric series allows all quadratic transformations for all series satisfying the corresponding applicability conditions to be obtained in the form of a small set of general basic relations. In other words any quadratic transformation for standard series, for example for the Gauss', Appell's, Horn's, Kampé de Fériet's, Lauricella's, Gelfand's series, etc., as well as for numerous nameless series, can be obtained as particular cases of the relations given in the paper. Along with completeness and generality of analysis the factorization method ensures an essential simplification of the theory by introducing a natural hierarchical structure into a system of quadratic connections between nine types of canonical forms. These results may contribute to development of advanced computer algebra systems capable to analyze, automatically, important properties of those multiple series which find large-scale applications in mathematics, mathematical physics and theoretical chemistry. 
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A. W. Niukkanen. Quadratic transformations of multiple hypergeometric series. Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 2, pp. 517-531. http://geodesic.mathdoc.fr/item/FPM_2002_8_2_a8/

[1] Niukkanen A. V., “Rasprostranenie printsipa faktorizatsii na gipergeometricheskie ryady obschego vida”, Mat. zametki, 67:4 (2000), 573–581 | MR | Zbl

[2] Niukkanen A. V., “Obschie lineinye preobrazovaniya gipergeometricheskikh funktsii”, Mat. zametki, 70:5 (2001), 769–779 | MR | Zbl

[3] Niukkanen A. V., Paramonova O. S., “Lineinye preobrazovaniya i formuly reduktsii gipergeometricheskikh funktsii Gelfanda, svyazannykh s grassmanianami $G_{2,4}$ i $G_{3,6}$”, Mat. zametki, 71:1 (2002), 88–89 | MR | Zbl

[4] Niukkanen A. V., “Formuly analiticheskogo prodolzheniya gipergeometricheskikh ryadov ot mnogikh peremennykh”, Fund. i prikl. mat., 7:1 (2001), 71–86 | MR | Zbl

[5] Niukkanen A. V., “Novaya teoriya gipergeometricheskikh ryadov i perspektivy ee ispolzovaniya v sistemakh kompyuternoi algebry”, Fund. i prikl. mat., 5:3 (1999), 717–745 | MR | Zbl

[6] Niukkanen A. W., “Operator factorization method and addition formulas for hypergeometric functions”, Integral transforms and special functions, 11 (2001), 25–48 | DOI | MR | Zbl

[7] Niukkanen A. V., “Novyi metod v teorii gipergeometricheskikh ryadov i spetsialnykh funktsii matematicheskoi fiziki”, Uspekhi mat. nauk, 43:3 (1988), 191–192 | MR

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

[9] Askey R., Orthogonal Polynomial and Special Functions, Philadelphia, Pa, 1975 | MR

[10] Niukkanen A. W., “Fourier transforms of atomic orbitals. I: Reduction to four-dimensional harmonics and quadratic transformations”, International Journal of Quantum Chemistry, 25 (1984), 941–955 | DOI | MR

[11] Appell P., Kampé de Fériet J., Fonctions hypergéométriques et hypersphériques: polynômes d'Hermite, Gauthier-Villars, Paris, 1926

[12] Srivastava H. M., Karlsson P. W., Multiple hypergeometric series, Ellis Horwood Series: Mathematics and its Applications, Ellis Hoorwood, Chichester, 1985 | MR | Zbl