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On partial differential equations leading to new classes of special functions. (English. Russian original) Zbl 0848.35027
Russ. Acad. Sci., Dokl., Math. 50, No. 1, 14-17 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 337, No. 1, 17-19 (1994).
Summary: In Yu. D. Pletner [Zh. Vychisl. Mat. Mat. Fiz. 30, No. 6, 868-882 (1990; Zbl 0706.35038); English transl. in USSR Comput. Math. Math. Phys. 30, No. 5, 161-171 (1990); Zh. Vychisl. Mat. Mat. Fiz. 32, 742-757 (1992); English transl. in USSR Comput. Math. Math. Phys. 32, No. 6, 645-658 (1992; Zbl 0798.35033); Zh. Vychisl. Mat. Mat. Fiz. 32, 890-903 (1992); English transl. in USSR Comput. Math. Math. Phys. 32, No. 6, 769-780 (1992; Zbl 0783.35009)], partial differential equations that are in some sense analogous to ordinary differential equations are studied. It turns out that one can construct their solutions using a fundamental system of solutions of the “associated” ordinary differential equations, in which case the properties of the “associated” solutions resemble those of the original to a large degree. In this connection, it is interesting to distinguish those classes of equations that are analogous to the hypergeometric equation, and thereby generate new classes of special functions. Such is our goal in this note.
The need to study these equations is dictated not just by purely mathematical interest, but also by the fact that they naturally arise in a series of problems in plasma physics and hydrodynamics that are of interest from an applied viewpoint (see loc. cit.).
##### MSC:
 35G05 Linear higher-order PDEs 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$