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i want to evaluate the following integral:

\begin{equation} a_n = \frac{2n+1}{2}\int_{-1}^1 \frac{P_n(x)}{\sqrt{2-2x}} \text{dx} \ \text{where $P_n(x)$ is the $n^{th}$ Legendre Polynomial. } \end{equation} I am expecting the result to be $1$. I tried to set it up on MAPLE and tried a large number of $n$'s ( 1,2 ,3,1000,2000) and the result is $1$. However, i am looking for a rigorous proof. I looked into the table of integrals involving legendre polynomials and could not find a case that suits mine.

Can anyone help me please?

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The generating function for Legendre polynomials is $$ \frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n}P_n(x)t^n$$

Just use $t=1$ in here to expand your integrand in Legendre polynomials and then use the orthogonality condition.

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  • $\begingroup$ Can you provide me the reference where we are allowed to replace $t$ by 1. All the references state that $|t|$ < 1. $\endgroup$ – outlawoutlawz Oct 11 '17 at 16:25
  • $\begingroup$ Why on earth was this downvoted? $\endgroup$ – Marcel Oct 23 '17 at 10:53
  • $\begingroup$ @Marcel: Probably because you do not state on what subsets that equality is valid (i.e. what are the allowed values for $x$ and $t$ in it). In particular, how does one know whether the power series in $t$ converges in $t=1$? What happens for $t=x=1$? $\endgroup$ – Alex M. Nov 4 '17 at 16:21
  • $\begingroup$ @AlexM. This is not intended as general theory, but only as an answer to the question. $\endgroup$ – Marcel Nov 4 '17 at 18:28
  • $\begingroup$ @Marcel: First, I'm not the downvoter. Second, even as an answer to the question it shouldn't ignore studying the matters of convergence that arise (and I see two of them: the convergence of the series and the convergence of some improper integrals at $1$). $\endgroup$ – Alex M. Nov 4 '17 at 19:28

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