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On the evaluation of an integral involving the Whittaker \(W\) function. (English) Zbl 1448.33010

Let \(W_{\kappa,\mu}(z)\) be the Whittaker function, which is a solution of the Kummer-type differential equation \[ w''+\left (-\frac{1}{4}+\frac{\kappa}{z}+\frac{\frac{1}{4}-\mu^2}{z^2}\right )w=0. \] The aim of this paper is the evaluation of the integral \[ I(\alpha,\beta)=\int_{\alpha}^{+\infty}W_{1,\beta}^2(x) \frac{dx}{x^2}, \] where \(\alpha\in(0,+\infty)\) and \(\beta\in \mathbb C\) are such that \(W_{1,\beta}(\alpha)=0.\) This type of integral appears in the analytic closed-form solution of a certain Sturm-Liouville problem.

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
34B24 Sturm-Liouville theory
39A10 Additive difference equations
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