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\begin{equation*} \int_{-a}^{a}e^{-\rho x}P_n \left( \frac{x}{a} \right) dx = (-1)^n\sqrt{\frac{2\pi a}{\rho}}J_{n+1/2}(a\rho) ~~~~~~\text{for $a>0$} \end{equation*}

I am just wondering this formula is correct or not? Thanks in advance!

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  • $\begingroup$ It does not look so, and actually the $a$ parameter is irrelevant since it can be removed through a substitution. An equivalent and simpler formulation is just to ask for the closed form of $$\int_{-1}^{1}e^{-\lambda x}P_n(x)\,dx. $$ $\endgroup$ – Jack D'Aurizio Nov 19 '17 at 8:26
  • $\begingroup$ Which by Rodrigues formula only depends on $$\int_{-1}^{1}(1-x^2)^n e^{-\lambda x}\,dx = B\left(\frac{1}{2},n+1\right)\cdot{}_0 F_1\left(n+\frac{3}{2},\frac{\lambda^2}{4}\right),$$ while Bessel functions are hypergeometric ${}_1 F_2$ functions. $\endgroup$ – Jack D'Aurizio Nov 19 '17 at 8:30
  • $\begingroup$ Anyway, it is enough to check if both sides fulfill (or don't) the same second-order differential equation. $\endgroup$ – Jack D'Aurizio Nov 19 '17 at 8:38

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