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I am trying a figure out Eigen-function expansion of the following kind. $$ \exp\left[{-\frac{1}{4\alpha} \left(x^2 + y^2\right)}\right] I_{l + \frac{1}{2}} \left(\frac{x y}{2 \alpha}\right) = \sum_{n} \frac{\phi_n(x) \phi_n(y)}{\lambda_n}\,, $$ where $I_n$ is modified Bessel function.

The expression suggests close connection with Hardy-Hille expansion. https://en.wikipedia.org/wiki/Laguerre_polynomials#Hardy%E2%80%93Hille_formula

However, the argument of the Bessel function seems to challenge a straightforward application of Hardy-Hille expansion. Any idea is welcome.

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    $\begingroup$ This also seems related to the generalized translation (associated to the Bessel differential operator) of a Gaussian. Perhaps that might give you something to look into. $\endgroup$
    – Cameron Williams
    Commented Apr 8 at 3:42
  • $\begingroup$ Can you please suggest me some reference? I am from Physics background and do not have training a like a Mathematician. $\endgroup$
    – Purnendu
    Commented Apr 8 at 15:27

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