Abstract
The paper is devoted to a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F 1 equation (equivalent to the Bessel equation), the Gegenbauer equation and the Hermite equation. In particular, recurrence relations of their solutions, their integral representations and discrete symmetries are discussed.
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Andrews G.E., Askey R., Roy R.: Special functions. Cambridge University Press, Cambridge (1999)
Dereziński, J., Majewski, P.: Conformal symmetries and hypergeometric type functions (in preparation)
Erdelyi A., Magnus W., Oberhettinger F., Tricomi F.G.: Higher Transcendental Functions, vols. I, II, III. McGraw-Hill, New York (1953)
Hochstadt H.: The Functions of Mathematical Physics. Wiley, New York (1971)
Infeld L., Hull T.: The factorization method. Revs Mod. Phys. 23, 21–68 (1951)
Vilenkin, N. Ja, Klimyk, A.U.: Representations of Lie Groups and Special Functions. vol 1. Kluwer, Dordrecht
Magnus W., Oberhettinger F., Soni R.: Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd edn. Springer, New York (1966)
Miller W.: Lie Theory and Special Functions. Academic Press, New York (1968)
Miller, W.: Symmetry and Separation of Variables. Addison-Wesley, Reading (1977)
Nikiforov U.: Special functions of mathematical physics. Birkhäuser, Basel (1988)
Olver, P.J.: Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics. Springer, New York (1986)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge
Rainville E.D.: Special Functions. The Macmillan Co., New York (1960)
Truesdell, C.: An essay toward a unified theory of special functions based upon the functional equation \({\partial F(z,\alpha)/{\partial}z=F(z,\alpha+1)}\). Annals of Mathematical Studies no. 18. Princeton University Press, Princeton (1948)
Vilenkin, N.Ya.: Special Functions and the Theory of Group Representations. Translations of Mathematical Monographs. AMS, Providence (1968)
Wawrzyńczyk, A.: Modern Theory of Special Functions. PWN, Warszawa (1978) (Polish)
Whittaker, E.T., Watson, G.N.: A course of Modern Analysis, vols. I, II, 4th edn. (reprint of 1927 ed.). Cambridge University Press, New York (1962)
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Communicated by Claude Alain Pillet.
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Dereziński, J. Hypergeometric Type Functions and Their Symmetries. Ann. Henri Poincaré 15, 1569–1653 (2014). https://doi.org/10.1007/s00023-013-0282-4
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DOI: https://doi.org/10.1007/s00023-013-0282-4