Comparison theorems for the $\nu$-zeroes of Legendre functions $P^ m_ \nu (z_ 0)$ when $-1<z_ 0<1$

Abstract:

We consider the problem of ordering the elements of $\{ \nu _j^m({z_0})\}$, the set of $\nu$-zeroes of Legendre functions $P_\nu ^m({z_0})$ for $m = 0,1, \ldots$ and ${z_0} \in ( - 1,1)$. In general, we seek to determine conditions on $(m,j)$ and $(n,i)$ under which we can assert that $\nu _j^m < \nu _i^n$. A number of such results were established in [2] for ${z_0} \in [0,1)$, and in the work that we present here we extend a number of these to the case ${z_0} \in ( - 1,1)$. In addition, we prove $\nu _{j + 1}^m < \nu _j^{m + 2}$ for ${z_0} \in ( - 1,0)$ and $\nu _2^3 < \nu _1^6$ for ${z_0} \in ( - 1,1)$. Using the results established here and in [2], we are able to determine the ordering of the first eleven $\nu$-zeroes of $P_\nu ^m({z_0})$ for $0 < {z_0} < 1$ and show that the twelfth $\nu$-zero is not necessarily distinct.