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?
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.