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Let $(L_n^{(\alpha)}(x))_n $ a sequence of Laguerre polynomials, for $n=0,1,..., $ and ${\alpha>-1}$, prove that : $$ n!L_n^{(\alpha)}(x)=x^{-\frac{\alpha}{2}}\int_0^{\infty}e^{x- y}y^{n+\frac{\alpha}{2}}J_{\alpha}(2\sqrt {xy})~dy,$$ where $$J_{\alpha}(t)=\sum_{k=0}^{\infty}\frac{(-1)^k(\frac{t}{2})^{2k+\alpha}} {k!\Gamma(k+\alpha+1)}$$ is the Bessel function of the first kind of order $\alpha$.

I tried to prove but I don't know how to connect the Laguerre polynomials explicit representation from the right-hand side of the integral.$\\$

Does anyone have any idea or proof of this?

metamorphy
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EBS
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1 Answers1

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A straightforward idea is to put the series for $J_\alpha$ directly into the integral (and exchange integration with summation, admissible because of absolute convergence). Recognizing an $n$-th derivative middleways, we reduce everything to the Rodrigues formula for $L_n^{(\alpha)}(x)$: \begin{align}\mathrm{RHS}&=x^{-\alpha}e^x\int_0^\infty y^n e^{-y}\sum_{k=0}^\infty\frac{(-1)^k(xy)^{k+\alpha}}{k!\Gamma(k+\alpha+1)}~dy\\&=x^{-\alpha}e^x\frac{\partial^n}{\partial x^n}\int_0^\infty e^{-y}\sum_{k=0}^\infty\frac{(-1)^k(xy)^{k+n+\alpha}}{k!\Gamma(k+n+\alpha+1)}~dy\\&=x^{-\alpha}e^x\frac{\partial^n}{\partial x^n}\sum_{k=0}^\infty\frac{(-1)^k x^{k+n+\alpha}}{k!\Gamma(k+n+\alpha+1)}\int_0^\infty y^{k+n+\alpha}e^{-y}~dy\\&=x^{-\alpha}e^x\frac{\partial^n}{\partial x^n}\left(x^{n+\alpha}\sum_{k=0}^\infty\frac{(-x)^k}{k!}\right)=\mathrm{LHS}.\end{align}

metamorphy
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