Prove

.

We can start as follows, by transforming it into a generalized hypergeometric function:

, since, from the series expansion of the generalized hypergeometric function, , where is the Pochhammer symbol .

Now the integrand function does not appear to be convergent numerically, except for where it becomes the Gaussian integral, and the case of where it becomes a Bessel function. For and , the integrand takes values of (serious). Beyond that the computer starts to produce smoke. Yet it eventually converges as there is a closed form solution. It is like saying that it works in theory but not in practice!

For, it turns out, under the restriction that , we can use the following result:

Allora, we can substitute , and with , given that ,

.

So either the integrand eventually converges, or I am doing something wrong, or both. Perhaps neither.

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*Related*

Maestro, I think that the identity follows from the Ramanujan’s master theorem, see https://en.wikipedia.org/wiki/Ramanujan%27s_master_theorem

Let ,

Mellin transform is ,

.

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Thanks a million!

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No, thank you, Maestro. I only learned about the theorem while trying to prove one of the previous identities, that you posted on Twitter some time ago.

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BTW, the paper at the end of that wiki article is worth reading, or at least skimming through.

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