We give an analytic solution of a second-order difference Poincaré-Perron-type equation. This enables us to construct a solution of the differential equation {fx1055-01} in explicit form. A solution of this equation is expressed in terms of two hypergeometric functions and one new special function. As a separate case, we obtain an explicit solution of the Heun equation and determine its polynomial solutions.
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References
H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York (1953).
E. Kamke, Differentialgleichungen Lösungsmethoden und Lösungen, Akademische Verlagsgesellschaft Geest, Leipzig (1959).
S. Slavyanov and W. Lay, Special Functions: A Unified Theory Based on Analysis of Singularities [in Russian], Nevskii Dialekt, St.Petersburg (2002).
G. Szegö, Orthogonal Polynomials [Russian translation], Fizmatgiz, Moscow (1962).
E. L. Ince, Ordinary Differential Equations [Russian translation], ONTI, Kharkov (1939).
V. V. Golubev, Lectures on the Analytic Theory of Differential Equations [in Russian], Gostekhizdat, Moscow (1950).
K. Ya. Latysheva and N. I. Tereshchenko, Lectures on the Analytic Theory of Differential Equations and Their Applications (Frobenius-Latysheva Method) [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1970).
K. Ya. Latysheva, N. I. Tereshchenko, and G. S. Orel, Normal-Regular Solutions and Their Applications [in Russian], Vyshcha Shkola, Kiev (1974).
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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 900–917, July, 2008.
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Kruglov, V.E. Solution of a second-order Poincaré-Perron-type equation and differential equations that can be reduced to it. Ukr Math J 60, 1055–1072 (2008). https://doi.org/10.1007/s11253-008-0120-x
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DOI: https://doi.org/10.1007/s11253-008-0120-x