Let $M_{\kappa,\mu}(z)$ be the Whittaker function, as defined here http://en.wikipedia.org/wiki/Whittaker_function.

Does any one know the evaluation of the following integral?

$$\int_{-\infty}^\infty \left|M_{i\alpha, \beta}\Big(\frac{i}{x}\Big)\right|^2dx,$$ where $\alpha \in \mathbb{R}$ and $\beta > 0.$

Any suggestion is welcome.

PS: I know in a Russian book: Integrals and series volume 3, there is a chapter on the integral of products of Whittaker functions, however, I can not find the book in the library of my university.

  • $\begingroup$ Here are its contents and screen of p. 255. $\endgroup$ – user64494 Sep 2 '14 at 6:39
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    $\begingroup$ It seems possible to evaluate the integral by the long calculation say in terms of the Meier G--function, but what for? What is your motivation? $\endgroup$ – Sergei Sep 7 '14 at 10:31
  • $\begingroup$ @Sergei Thank you. My motivation is the following: these Whittaker functions appear in the scaling limit of some orthogonal polynomials. I already know the norms of the orthogonal polynomials and the limit of the norms. So I want to know the norm of the limit function, if the norm of the limit function coincides with the limit of the norms, then I can conclude that the convergence is in $L^2$ sense, this is what I am looking for. By the way, if $\alpha =0$, this reduced to function related to Bessel functions and the norm is in fact known. Fortunately, this is sufficient for me. $\endgroup$ – Yanqi QIU Sep 8 '14 at 11:43

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