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.



Estimating associated Legendre function
I randomly (luckily) found this link <http://dlmf.nist.gov/14.26> just now. I hope I can find desired answer from here. Thanks for viewing my question and still, I will appreciate any advise.


