Ordering the zeroes of Legendre functions \(P^ m_{\nu}(z_ 0)\) when considered as a function of \(\nu\)
Let m denote a nonnegative integer and let \(P^ m_{\nu}(z)\) denote the associated Legendre functions of the first kind; i.e. \(P^ m_{\nu}(z)\) is a solution of the boundary value problem \[ ((1-z^ 2)y'(z))'+(\nu (\nu +1)-\frac{m^ 2}{1-z^ 2})y(z)=0, \] y(1) is bounded. For \(z_ 0\in (0,1)\), let \(\nu^ m_ j=\nu^ m_ j(z_ 0)\) denote the jth positive root of \(P^ m_{\nu}(z_ 0)=0\). By standard Sturm-Liouville techniques, it is not difficult to compare successive positive roots of \(P^ m_{\nu}(z_ 0)=0\) and \(P_{\nu}^{m+1}(z_ 0)=0\) and obtain \[ (1)\quad \nu^ m_ j(z_ 0)<\nu_ j^{m+1}(z_ 0)<\nu^ m_{j+1}(z_ 0)\quad (j=1,2,...). \] However, a natural question to ask is: \[ (2)\quad how\quad does\quad \nu^ m_ j(z_ 0)\quad compares\quad to\quad \nu^ n_ i(z_ 0)? \] The author proves the following inequalities: \(\nu_ j^{m+2}(z_ 0)<\nu^ m_{j+1}(z_ 0)\), \(\nu^ 0_ 2(z_ 0)<\nu^ 3_ 1(z_ 0)\), \(\nu^ 1_ 2(z_ 0)<\nu^ 4_ 1(z_ 0)\), and \(\nu^ 0_ 3(z_ 0)<\nu^ 5_ 1(z_ 0)\), all valid for \(z_ 0\in (0,1)\). By combining these inequalities with (1), we have \[ \nu^ 0_ 1<\nu^ 1_ 1<\nu^ 2_ 1<\nu^ 0_ 2<\nu^ 3_ 1<\nu^ 1_ 2<\nu^ 4_ 1<\nu^ 2_ 2<\nu^ 0_ 3<\nu^ 5_ 1, \] valid for all \(z_ 0\in (0,1)\). This provides a partial answer to the question posed in (2).
scientific article; zbMATH DE number 4208508