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I need to numerically evaluate the following integral $$\sqrt{\frac{n!(n+1)!}{(n+\alpha)!(n+1+\alpha)!}}\int_0^\infty \frac{1}{\sqrt{x+c}}x^\alpha e^{-x}L_n^\alpha(x)L_{n+1}^\alpha(x)\;\mathrm dx$$

where $c\ge 0$, and $n,\alpha$ are integers. I have no problem in evaluating this directly with available numerical packages in cases where $\alpha$ and $n$ are small. However, at large $n$ or $\alpha$ (say larger than ~100), the integrand becomes highly oscillatory, and even worse, the values of the Laguerre polynomial itself are no longer reliable (according the documentation of the package I'm using).

So, is there a method to simplify the integrand to mitigate these problems?

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MMa
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    I'm afraid "some function $f(x)$" is a bit too vague. Could you be more specific, please? –  Nov 03 '20 at 11:48
  • I edited the question. I was hoping there was a method that would work with an arbitrary "well behaved" function $f$. Hopefully now the question is more meaningful. – MMa Nov 03 '20 at 19:47
  • Does your notation $(n+\alpha)!$ mean that $\alpha$ is an integer, or do you use it just as a synonym for $\Gamma(n+\alpha+1)$? –  Nov 03 '20 at 20:48
  • @ProfessorVector Yes, $\alpha$ is an integer. – MMa Nov 03 '20 at 21:11
  • Have you tried a (generalized) Laguerre expansion$$\frac1{\sqrt{x+c}}\sim\sum^\infty_{n=0}a_n,L^\alpha_n(x)?$$ You don't need pointwise convergence to evaluate integrals. $L_n^\alpha(x)L_{n+1}^\alpha(x)$ is just a polynomial and can be written as a linear combination of $L_m^\alpha(x)$ for $m=0,\ldots,2n+1$, and then, you use orthogonality.If that process is numerically stable, is another question, though... –  Nov 04 '20 at 08:10

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