(And a related question: Where should an analytic number theorist learn about Bessel functions?)

Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of Iwaniec and Kowalski, says the following. Let $r(m)$ be the number of representations of $m$ as two squares, and suppose that $g$ is smooth and compactly supported in $(0, \infty)$. Then,

$$\sum_{m = 1}^{\infty} r(m) g(m) = \pi \int_0^{\infty} g(x) dx + \sum_{m = 1}^{\infty} r(m) h(m),$$

where $$h(y) = \pi \int_0^{\infty} g(x) J_0(2 \pi \sqrt{xy}) dx.$$

$J_0(x)$ is a Bessel function, and I+K follow with four equivalent integral expressions -- the equivalence of which is not at all obvious by looking at them. Looking at Lemma 4.17, the relevance appears to be that Bessel functions arise when you take Fourier transforms of radially symmetric functions.

Another example comes from (3.8) of this paper of Miller and Schmid, and the relevance comes from the identity

$$\int_0^{\infty} J_0(\sqrt{x}) x^{s - 1} dx = 4^s \frac{\Gamma(s)}{\Gamma(1 - s)},$$

where the gamma factors come from functional equations of $L$-functions. Okay, if this is true, then I understand why we care, but it seemed a bit *deus ex machina* to me.

There are many other examples too, for example the Petersson formula in the theory of modular forms, etc. There are $I$-Bessel functions, $K$-Bessel functions, $Y$-Bessel functions, etc., all of which seem to satisfy a dizzying number of highly nontrivial identities, and reading Iwaniec and Kowalski one gets the sense that an expert should have the ability to recognize and manipulate them on sight. They also provide references to, e.g., (23.451.1) of a book by Gradhsteyn and Rizhik, and although I confess I have not looked at it, I can infer from the formula number that it is not the sort of thing I might read on an airport layover.

Meanwhile, Wikipedia tells me that they naturally arise as solutions of certain partial differential equations. Looks extremely interesting, although I'm afraid I am not an expert in PDE.

As an analytic number theorist, how might I make friends with these objects? How should I look at them, and what conceptual frameworks do they fit in? Thank you!

(ed. Thanks to everyone for informative answers! I could only accept one answer but +1 all around)

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