I have the following identity (Bateman vol. 3, 19.10.2): \begin{align} \left(1-2tx+t^2\right)^{-\frac{\ell+1}{2}}P_\ell^{(m)}\left(\frac{x-t}{\sqrt{1-2tx+t^2}}\right)=\sum_{k=0}^\infty\binom{\ell-m+k}{k}P_{\ell+k}^{(m)}(x)t^k, \end{align} where $\ell$ and $m$ are integers with $|m| \leq \ell$ and $P_\ell^{(m)}(x)$ is the associated Legendre function. I would like to show that for $|t| < 1$, the sum on the right hand side converges uniformly in $x$ for $|x| \leq 1$.
For any fixed $x\in [-1,1]$, the right hand side converges pointwise to the left hand side for $|t| < 1$. It is also a power series in $t$, which means that it converges uniformly on any closed subinterval of the interval of convergence, which is $(-1,1)$.
However, I want to show uniform convergence in $x$, not just in $t$. More formally, let $f(x,t)$ be the left hand side and let $f_n(x,t)$ be the $n$th partial sum of the right hand side. If $|x| \leq 1$, then for any given $\epsilon$, there exists an $N(x,\epsilon)$ such that \begin{align} \left|f(x,t) - f_n(x,t)\right| < \epsilon, \end{align} for all $n \geq N(x,\epsilon)$ and for all $t \in (-1,1)$. The uniform convergence as a sequence of functions in $t$ ensures that $N(x,\epsilon)$ is independent of $t$. Is there a way to show that it is also independent in $x$?
Bonus question: More generally, if I have generating function $G(x,t)$ for some set of functions $P_k(x)$, i.e., \begin{align} G(x,t) = \sum_{k=0}^\infty P_k(x) t^k, \end{align} is there a generic way to rewrite this as \begin{align} G(x,t) = \sum_{k=0}^\infty Q_k(t) x^k, \end{align} for some new functions $Q_k(t)$? If there's a procedure to do this, then I think the uniform convergence in $x$ I want above follows automatically.
