Geodesic
On the theory of special functions and their approximations
Sbornik. Mathematics, Tome 59 (1988) no. 2, pp. 429-458 
V. K. Dzyadyk. On the theory of special functions and their approximations. Sbornik. Mathematics, Tome 59 (1988) no. 2, pp. 429-458. http://geodesic.mathdoc.fr/item/SM_1988_59_2_a10/
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     url = {http://geodesic.mathdoc.fr/item/SM_1988_59_2_a10/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

For general nonhomogeneous linear equations with one regular singular point, the questions of solvability, the structure of the corresponding Wronskians, and the Cauchy–Green functions are studied on a finite interval. Formulas for the resolvents of the resulting integral equations are obtained in explicit form. An effective method of constructing differential equations for the analytic parts of special functions is indicated, and an analytical method is worked out for their best asymptotic approximation, which at the same time gives a high degree of accuracy in their practical computation on computers. Bibliography: 17 titles.

[1] Dzyadyk V. K., “Approksimatsionnyi metod priblizheniya algebraicheskimi mnogochlenami reshenii lineinykh differentsialnykh uravnenii”, Izv. AN SSSR. Ser. matem., 38 (1974), 937–967 | Zbl

[2] Stolyarchuk V. K., “O ravnomernom priblizhenii funktsii Besselya s tselym indeksom mnogochlenami”, Ukr. matem. zhurn., 26:5 (1974), 670–674

[3] Stolyarchuk V. K., “O postroenii dlya funktsii $\operatorname{si}x$ i $\Phi(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^2}\,dt$ mnogochlenov, kotorye osuschestvlyayut ikh priblizhenie, blizkoe k nailuchshemu”, Ukr. matem. zhurn., 26:2 (1974), 215–224

[4] Stolyarchuk V. K., “O ravnomernom priblizhenii mnogochlenami na segmente gipergeometricheskikh funktsii”, Voprosy teorii priblizheniya funktsii i ee prilozhenii, Kiev, 1976, 179–185

[5] Gelfond A. O., Ischislenie konechnykh raznostei, Nauka, M., 1967 | MR

[6] Khartman F., Obyknovennye differentsialnye uravneniya, Mir, M., 1970 | MR | Zbl

[7] Gursa E., Kurs matematicheskogo analiza, t. II, M.-L., 1936

[8] Lyuk Yu., Spetsialnye matematicheskie funktsii i ikh approksimatsii, Mir, M., 1980

[9] Abramovits M., Stigan I., Spravochnik po spetsialnym funktsiyam, Nauka, M., 1979 | MR

[10] Aigner M., Kombinatornaya teoriya, Mir, M., 1982 | MR

[11] Fedoryuk M. F., Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, M., 1983 | MR | Zbl

[12] Koddington E. A., Levinson N., Teoriya obyknovennykh differentsialnykh uravnenii, IL, M., 1958

[13] Markov A. A., Izbrannye trudy, M.-L., 1948

[14] Bernshtein S. N., Ob odnom klasse ortogonalnykh mnogochlenov, Sobr. soch., t. 1, 1952, 452–467

[15] Akhiezer N. I., Lektsii po teorii approksimatsii, Nauka, M., 1966, 290–294 | Zbl

[16] Dzyadyk V. K., “On a problem of Chebyshev and Markov”, Analysis Math., 3.3 (1977), 171–175 | DOI | Zbl

[17] Pashkovskii S., Vychislitelnye primeneniya mnogochlenov i ryadov Chebysheva, Nauka, M., 1983 | MR