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Some expansions of arbitrary functions in integrals with respect to functions of a parabolic cylinder. (Russian) Zbl 0783.33005
The author studies expansions of functions in integrals with respect to some combinations of functions of a parabolic cylinder \[ \begin{split} P_{1,2}(x,\xi)= \{\Gamma(\gamma_{2,1}-in^ 2 /4x^ 2) [H_{- 1/2+in^ 2/2x^ 2} (\xi xe^{i\pi/4})\mp H_{-i/2+in^ 2 /2x^ 2} (-\xi xe^{2\pi/4})]\}/\\ \sqrt{\pi}\cdot 2^{1/2+in^ 2/2x^ 2} \exp[ix^ 2 \xi^ 2/2+ i\pi(\gamma_{1,2}-1/4)2]=\\ \exp(-ix^ 2 \xi^ 2/2) F(\gamma_{1,2}-in^ 2/4x^ 2,2\gamma_{1,2} ix^ 2\xi^ 2) {\begin{cases} 2x\xi,\\ 1,\end{cases}} \quad \gamma_ 1=3/4, \quad \gamma_ 2=1/4,\end{split} \] where \(H_ \nu(z)\) is the Hermite function, \(P(z)\) is the gamma function and \(F(\alpha,\beta,z)\) is the degenerate hypergeometric function.
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C99 Nontrigonometric harmonic analysis