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Les fonctions hypercylindriques dans l’espace à $n + 2$ dimensions. (French) JFM 47.0348.02
Setzt man im vierdimensionalen Raume $$x=\varrho \sin\theta \sin \psi,\ y =\varrho \sin \theta \cos \psi,\ z=\varrho \cos\theta, \ t=t,$$ und sucht harmonische Funktionen von der Form $$U = e^{\mu t}\cos \nu \psi V(\varrho, \theta)$$ ($\mu, \nu$ Konstant), so kommt man auf die vom Verf. betrachteten Hyper-Zylinderfunktionen $V.$ Sie genügen der Differentialgleichung $$\varrho^2\frac {\partial^2V}{\partial \varrho^2} +(1- \omega^2)\frac {\partial^2V}{\partial \omega^2} +2\varrho\frac {\partial V}{\partial \varrho}-2\omega\frac {\partial V}{\partial \omega} +\mu^2\varrho^2V- \frac{\nu^2}{1-\omega^2}V =0,$$ wenn $\cos \theta = \omega$ gesetzt wird. Sie stehen in einfacher Beziehung zu den Appelschen hypergeometrischen Funktionen zweier Veränderlichen. Die zweite Note überträgt diese Betrachtungen auf höhere Räume. (IV 13.)
Reviewer: Szegö, Prof. (Berlin)