For past months I've been trying to estimate associated Legendre function $P_{-\frac{1}{2}+it}^m (\cosh r)$ in order to study Laplacian eigenfunctions on a hyperbolic surface. I found reasonably sharp upper bound of $C_m(t,r):=m!P_{-\frac{1}{2}+it}^{-m} (\cosh r)$ for non-negative $m$ when,
1) $r$ is fixed, $t$ is large, and $m$ is arbitrary,
2) $t,m$ are fixed,
3) or $r,t$ are fixed.
I believe I can deduce meaningful bound when $m$ is relatively larger than $t$ and $r$. (When only $t$ is fixed, the estimation I found doesn't seem sharp enough.)
However, non of these estimations I found led me to estimation of $C_m(t,r)$ which is sharp and uniform in $m,t,r$. I'm mainly interested in finding uniform upper bound for $C_m(t,r)$ which coincides with 1),2),3).
If there's no known result regarding this question, I would like to bound $\sup_{r \in \mathbb{R}} C_m(t,r)$ in terms of $t,m$, and see for which $r$ the supremum being achieved.
Can anyone help me on these? Any suggestion or advise on these will be welcomed.
