1
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ Here are its contents and screen of p. 255. $\endgroup$
    – user64494
    Sep 2, 2014 at 6:39
  • 1
    $\begingroup$ 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, 2014 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, 2014 at 11:43

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.