On common zeros of Legendre’s associated functions

Abstract:

In this paper it is proved that any two given Legendre associated functions $P_n^m(\mu )$ and $P_n^s(\mu )$, where $n \geqslant 1$ is an integer and where one of the integers m or s may be 0 (and $m \ne \pm s$), have either no zero in common or exactly one common zero, namely $\mu = 0$. An auxiliary result states that the $n - m$ zeros of $P_n^m$ known to lie in the open interval $( - 1,1)$ lie in fact in the open interval $( - c,c)$, where $\pm c$ are the two zeros of $n(n + 1) - {m^2}/(1 - {\mu ^2})$ which is one of the coefficients in the Legendre associated equation satisfied by $P_n^m$. Some monotonicity behavior of $P_n^m$ is simultaneously described. The proof of the main result is based on properties of Prüfer polar coordinates.