A recurrence algorithm for finding particular solutions of the resonant equation of the fourth order connected with the generalization of Laguerre and Legendre polynomials is constructed and substantiated. For this purpose, we use the general theorem on the representation of particular solutions of resonant equations in Banach spaces proved by Makarov in 1976. An example of the general solution of resonant equations with differential operator for the Laguerre-type polynomials is presented.
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A. P. Abdullaev and A. B. Burmistrova, “On the one scheme of investigation of the solvability of resonant boundary-value problems,” Izv. Akad. Nauk SSSR, Ser. Mat., No. 11, 14–22 (1996); English translation: Russian Math. (Izv. Vuzov), 40, No. 11, 12–20 (1996).
N. B. Backhouse, “Resonant equations and special functions,” J. Comput. Appl. Math., 133, No. 1-2, 163–169 (2001); https://doi.org/10.1016/S0377-0427(00)00641-5.
N. B. Backhouse, “The resonant Legendre equation,” J. Math. Anal. Appl., 117, No. 2, 310–317 (1986); https://doi.org/10.1016/0022-247X(86)90227-1.
V. L. Makarov, Orthogonal Polynomials and Difference Schemes with Exact and Explicit Spectra [in Russian], Doctoral-Degree Thesis (Physics and Mathematics), Kiev (1976).
V. L. Makarov and T. Arazmyradov, “On the construction of particular solutions of resonant differential equations,” Differents. Uravn., 14, No. 7, 1255–1261 (1978).
V. L. Makarov, “FD-method: The exponential rate of convergence,” Zh. Obchysl. Prykl. Mat., No. 82, 69–74 (1997); English translation: J. Math. Sci., 104, No. 6, 1648–1653 (2001); https://doi.org/10.1023/A:1011333314231.
V. L. Makarov and N. M. Romaniuk, “Exponentially convergent symbolic algorithm of the functional-discrete method for the fourth order Sturm–Liouville problems with polynomial coefficients,” J. Comput. Appl. Math., 358, 405–423 (2019); https://doi.org/10.1016/j.cam.2019.03.024.
I. Gavrilyuk and V. Makarov, “The classical orthogonal polynomials in resonant equations. I,” J. Numer. Appl. Math., 130, No. 1, 58–70 (2019).
I. Gavrilyuk and V. Makarov, “Resonant equations with classical orthogonal polynomials. I,” Ukr. Mat. Zh., 71, No 2, 190–209 (2019); Ukr. Math. J., 71, No 2, 215–236 (2019).
I. Gavrilyuk and V. Makarov, “Resonant equations with classical orthogonal polynomials. II,” Ukr. Mat. Zh., 71, No 4, 455–470 (2019); Ukr. Math. J., 71, No 4, 519–536 (2019).
A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics [in Russian], Nauka, Moscow (1978).
A. M. Krall, “Orthogonal polynomials satisfying fourth order differential equations,” Proc. Roy. Soc. Edinburgh. Sect. A, 87, No. 3-4, 271–288 (1981); https://doi.org/10.1017/S0308210500015213.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 11, pp. 1529–1538, November, 2019.
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Makarov, V.L., Romaniuk, N.M. & Bandyrskii, B.I. Generalization of Resonant Equations for the Laguerre- and Legendre-Type Polynomials to Equations of the Fourth Order. Ukr Math J 71, 1751–1762 (2020). https://doi.org/10.1007/s11253-020-01745-6
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DOI: https://doi.org/10.1007/s11253-020-01745-6