Inequalities for the associated Legendre functions
The main results are about the bound for the associated Legendre functions of the first kind \(P_n^m(x)\) with fixed \(x\in [-1,1]\) and integer \(m,n,1\leq | m| \leq n.\) First of all, the author proves a result about transferring upper bounds of the Legendre polynomials \(P_n(x)\) to the functions \(P_n^m(x).\) That result allows to obtain bounds of the type \[ A(\alpha ,n,m)\leq \max _{x\in [-1,1]} | (1-x^2)^{\alpha/2}P_n^m(x)| \leq B(\alpha ,n,m) \] for all \(0\leq \alpha \leq 1/2\) and \(1\leq | m| \leq n.\) For \(\alpha =0\) and \(\alpha =1/2\) these upper bounds are improvements and simplifications of known results.
scientific article; zbMATH DE number 1297475