Geodesic
Structural theory of special functions
Teoretičeskaâ i matematičeskaâ fizika, Tome 119 (1999) no. 1, pp. 3-19 Cet article a éte moissonné depuis la source Math-Net.Ru

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A block diagram is suggested for classifying differential equations whose solutions are special functions of mathematical physics. Three classes of these equations are identified: the hypergeometric, Heun, and Painlevé classes. The constituent types of equations are listed for each class. The confluence processes that transform one type into another are described. The interrelations between the equations belonging to different classes are indicated. For example, the Painlevé-class equations are equations of classical motion for Hamiltonians corresponding to Heun-class equations, and linearizing the Painlevé-class equations leads to hypergeometric-class equations. The “confluence principle” is stated, and an example of its application is given. 
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S. Yu. Slavyanov. Structural theory of special functions. Teoretičeskaâ i matematičeskaâ fizika, Tome 119 (1999) no. 1, pp. 3-19. http://geodesic.mathdoc.fr/item/TMF_1999_119_1_a0/

[1] F. Olver, Asimptotika i spetsialnye funktsii, Nauka, M., 1990 | MR | Zbl

[2] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, V trekh tomakh, 1965 ; 1966; Наука, М., 1967 | MR

[3] E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1959 | MR

[4] E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1927 | MR | Zbl

[5] P. Moon, D. E. Spencer, Field Theory Handbook, Springer, Berlin–Heidelberg–New York, 1971 | MR

[6] K. Iwasaki, H. Kimura, S. Shimomura, M. Ioshida, From Gauss to Painleve: a Modern Theory of Special Functions, Vieweg, Braunschweig, 1991 | MR | Zbl

[7] R. G. Barantsev, “Definitsiya asimptotiki i sistemnye triady”, Asimptoticheskie metody v teorii sistem, ed. A. N. Panchenkov, SO AN SSSR, Irkutsk, 1980, 70–81 | MR

[8] A. Zeeger, V. Lai, S. Yu. Slavyanov, TMF, 104:2 (1995), 233–247 | MR

[9] Centennial Workshop on Heun's Equation – Theory and Applications, eds. A. Seeger, W. Lay, MPI/MF, Stuttgart, 1990

[10] A. Decarreau, M. C. Dumont-Lepage, P. Maroni, A. Robert and A. Ronveaux, Ann. Soc. Sc. Bruxelles, 92 (1978), 53–78 | MR | Zbl

[11] A. Decarreau, P. Maroni and A. Robert, Ann. Soc. Sc. Bruxelles, 92 (1978), 151–189 | MR | Zbl

[12] A. Ronveaux (ed.), The Heun's Differential Equation, Oxford University Press, Oxford, 1995

[13] S. Yu. Slavyanov, J. Phys. A, 29 (1996), 7329–7335 | DOI | MR | Zbl