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On solution of equation, arising in the Dirichlet problem for ultrahyperbolic equation in a ball. (Russian) Zbl 1137.33316

Let \(\Omega\) be a unit ball in \(R^n\), \(0<k<n\), \(a\in C\), \(I_i=-i(i+k+2)\), \(J_j=-j(j+n-k-2)\). The investigation of the homogeneous Dirichlet problem for ultrahyperbolic equation \[ u_{x_1x_1}+...+u_{x_kx_k} -a^2\left(u_{x_{k+1}x_{k+1}}+\dots+u_{x_nx_n}\right) =0,\,\, in \,\, \Omega;\quad u|_{\partial\Omega}=0 \] leads to the following equation \[ \begin{split} u_{m+2}''(\phi)+((n+k-1) \cot \phi-(k-1)\tan)u'_{m+2}(\phi)+((m+2)(m+n)+\\ +I_i\cos(\phi)^{-2}+J_j\sin(\phi)^{-2})u_{m+2}(\phi)=0,\end{split}\tag{1} \] and relations \[ u_{m+2}(\phi)|_{\cot(\phi)=\pm\alpha}=0.\tag{2} \] The solution of (1), (2) is constructed in the form of the \((m+2)\)-order polynomial with respect to the \(\cos(\phi)\) and \(\sin(\phi)\).

MSC:

33D90 Applications of basic hypergeometric functions
34B08 Parameter dependent boundary value problems for ordinary differential equations
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