# zbMATH — the first resource for mathematics

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.
##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 42C99 Nontrigonometric harmonic analysis
Hermite function