Abstract
In a number of cases pairs of Goursat’s higher-order hypergeometric transformations lend themselves to eliminations of the right-hand members; one just has to make his or her variables equal by solving certain algebraic equations of degree up to six. This is carried out by the aid of Maple, in most cases supplemented by suitable one-to-one variable substitutions.
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References
Berndt, B.C., Bhargava, S., Garvan, F.G.: Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347, 4163–4244 (1995)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. I. McGraw-Hill, New York (1953)
Garvan, F.G.: Ramanujan’s theories of elliptic functions to alternative bases – a symbolic excursion. J. Symbolic Comput. 20, 517–536 (1995)
Goursat, É.: Sur l’équation différentielle linéaire qui admet pour intégrale la série hypergéométrique. Ann. Sci. École Norm. Sup. 10(2), S3–S142 (1881)
Karlsson, P.W.: On some hypergeometric transformations. Panam. Math. J. 10(4), 59–69 (2000)
Kummer, E.E.: Über die hypergeometrische Reihe…. Crelles J. 15, 39–83, 127–172 (1836)
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Karlsson, P.W. (2014). Goursat’s Hypergeometric Transformations, Revisited. In: Milovanović, G., Rassias, M. (eds) Analytic Number Theory, Approximation Theory, and Special Functions. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0258-3_27
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DOI: https://doi.org/10.1007/978-1-4939-0258-3_27
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