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This is pretty much just a yes or no question (an explanation on top would probably help though).

I'm asked to find the derivative of the following generating function:

$$ G(x,r) = \sum_{n=0}^\infty P_n(x) r^n = (1 - 2rx + r^2)^{-\frac{1}{2}} \quad \quad |x| \leq 1, \ |r| < 1.$$

I asked a question on integrating this function previously: Re-arranging and integrating a Generating Function of Legendre Polynomials

My main confusion on this question was that I didn't know that we didn't have any dependency on $r$ so we could just ignore it.

My question is; does the same apply for finding the derivative? Do I ignore the $r$ terms when finding the derivative or keep them in?

I've calculated the derivative with the $r$ terms included as $r/{(1-r(2x-r))}^{3/2}$ but this doesn't look quite right so I'm leaning more towards just ignoring the $r$ terms, could anyone clarify what I should do?

Thanks in advance

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  • $\begingroup$ I've edited your question to use mathjax rather than an image so that future users with similar questions will have an easier time searching for this question. In the future you should do the same ^_^ $\endgroup$ May 25, 2021 at 14:53
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    $\begingroup$ Ok, thanks, to be honest I was just being a bit lazy but next time I'll write it out, it's probably better to. $\endgroup$
    – Charlie P
    May 25, 2021 at 15:01

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