Harmonic functions vanishing on cones: an existence criterion (Q1577182)

The author presents a complete solution of the following problem stated in the survey of \textit{K. B. Barth, D. A. Brannan}, and \textit{W. K. Hayman} [Bull. Lond. Math. Soc. 16, 490-517 (1984; Zbl 0593.30001)]: For what \(\alpha\) is there a nonzero harmonic function in \(\mathbb R ^n\) (\(n \geq 3\)) vanishing on the cone \(K_{\alpha} = \{ (x _1, x') \in \mathbb R^n : |x'|= \alpha x _1\}\)? The main theorem states that for existence of a nonzero harmonic function in \(\mathbb R ^n\) (\(n \geq 3\)) vanishing on the cone \(K _\alpha\) it is necessary and sufficient that there exists a Gegenbauer polynomial in the sequence \(P^{\beta + i}_{k-i} (\cos \vartheta_1)\), where \(\beta = \frac{n-2}{2}\), \(i=0,1,\dots,k\), \(k=0,1,2,\dots\), \(\tan (\vartheta_1) = \alpha\), for which \((1 + \alpha^2)^{-1/2}\) is zero.