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(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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You might want to look at Siegel's work on transcendental numbers. –  Franz Lemmermeyer Aug 30 '12 at 17:16
My impression is that many families of special functions fit naturally into the setting of representation theory; they are matrix coefficients of suitable representations of suitable groups. A simple example is the trigonometric functions, which are matrix coefficients of $S^1$. Bessel functions are apparently related in this way to $\text{Isom}(\mathbb{R}^2)$. (But I don't know how much analytic number theorists like representation theory.) –  Qiaochu Yuan Aug 30 '12 at 17:21
Zach Kent once told me that he had to figure this out for himself as part of his thesis. Maybe get in contact with him? –  stankewicz Aug 30 '12 at 17:54
Actually, analytic number theorists like representation theory a lot! For example, modular forms can be described in terms of representations of $GL_2$ on certain function spaces. –  Frank Thorne Aug 30 '12 at 18:32
Bessel functions appear as coefficients in series expansions of automorphic forms on $GL_2(K)$, where $K$ is an imaginary quadratic field, in much the same way that exponential functions appear in the $q$-expansions of modular forms. This is explained rather nicely in Shai Haran's 1987 Compositio paper on p-adic L-functions. (This comment is more or less a special case of Qiaochu's comment.) –  David Loeffler Aug 30 '12 at 18:50
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7 Answers

up vote 21 down vote accepted

Radial Fourier transforms provide a good, consistent perspective on most of the theory. The Fourier transform $\widehat{f}(t)$ of a function $f \colon \mathbb{R}^n \to \mathbb{R}$ is given by the integral of $f(x) e^{2\pi i \langle x,t \rangle} \, dx$ over $x \in \mathbb{R}^n$. If $f$ is a radial function (i.e., $f(x)$ depends only on $|x|$), then we can radial symmetrize everything and the exponential function averages out to a radial function. Specifically, we get $$ \widehat{f}(t) = 2\pi |t|^{-(n/2-1)} \int_0^\infty f(r) J_{n/2-1} (2 \pi r |t|) r^{n/2} \, dr. $$ The precise factors are a little annoying, but basically this just means $J_{n/2-1}$ is what you get when you radially symmetrize an exponential function in $n$ dimensions. It's easy to see that if you symmetrize $e^{2\pi i \langle x,t \rangle}$ by averaging over all $x$ on a sphere, then you get a radial function of $t$, and furthermore as you vary the radius of the sphere you just rescale the function. So the one function $J_{n/2-1}$ captures all of this, modulo scaling.

One consequence is that Bessel functions inherit the orthogonality of the exponential functions (i.e., the different scalings are orthogonal), so they also inherit all the consequences of orthogonality. For example, this is really where the differential equation comes from. There's a strong analogy between Bessel functions and orthogonal polynomials, where rescaling the Bessel function corresponds to varying the degree of the polynomial.

You also get certain qualitative results for free: for example, the product of two Bessel functions should be an integral of Bessel functions with positive coefficients, since this corresponds to saying the product of two radial, positive-definite functions remains positive definite. You can write down the coefficients explicitly, but sometimes all you need is nonnegativity, and in any case this point of view makes it easy to believe that there should be an explicit formula.

This is basically a low-brow version of the representation theory approach. Basically, ordinary Fourier analysis studies $L^2(\mathbb{R}^n)$ under the action of the translation group $\mathbb{R}^n$. If you look at the full group of isometries of $\mathbb{R}^n$ (including the orthogonal group), then it's just a little more elaborate, and the Bessel functions arise as zonal spherical functions. It's worthwhile working through this perspective, but in practice just thinking about radial Fourier analysis gives you most of the benefits with less machinery.

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Isn't there some $f$ missing in the formula for $\hat f$? –  Dirk Aug 30 '12 at 20:42
Oops, thanks! I just added it. –  Henry Cohn Aug 30 '12 at 20:56
I like this, of course, though it still does not account for Bessel functions whose order is not in $\frac12\bf Z$. –  Noam D. Elkies Aug 31 '12 at 0:09
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There are many, many things that can be said here! I think there are two slightly different mechanisms that conjure up Bessel functions of various types, namely, Euclidean Laplacians and separation of variables, and $SL_2(\mathbb R)$ (and orthogonal groups $O(n,1)$) Casimir or Laplace-Beltrami operator and separation of variables. The fact that there cannot be a greater variety of second-order ordinary differential equations with certain control and types of singular points goes back to Riemann.

A very tangible connection with down-to-earth examples from automorphic forms: just as the function $z\rightarrow y^s$ is the spherical vector in an unramified principal series, the integral $\int_{-\infty}^\infty e^{-ix} {y^s\over |cz+d|^{2s}}\;dx$ that is the starting point for integral expressions for $K$-type Bessel functions is the image of the spherical vector $y^s$ under the obvious (from the viewpoint of "Mackey theory") intertwining from that principal series to the Whittaker space.

Not only in the Euclidean case, but also generally, very many "formulas" are manifestations either of a provably unique intertwining operator (given by an integral), or of a Plancherel theorem for the situation at hand.

(The fact that Mellin transforms of Bessel functions (for the Fourier expansions of waveforms) are expressible in terms of Gamma is not typical of integral transform methods on larger groups, unfortunately. The classical computations of archimedean integrals, for Rankin-Selberg, etc., for $SL_2(\mathbb R)$ are done in a self-contained fashion in http://www.math.umn.edu/~garrett/m/v/standard_integrals.pdf, for example.)

The asymptotics are more systematically understandable from the more general results on asymptotics of solutions of second-order ordinary differential equations. The regular singular point theory is very old, and even the good-irregular singular point case has been essentially understood since Poincare. It turns out that fairly simple heuristics are provably correct, and therefore function as excellent mnemonics. (I wrote up some examples and proofs in a more contemporary style in some course notes: http://www.math.umn.edu/~garrett/m/mfms/notes_c/reg_sing_pt.pdf, http://www.math.umn.edu/~garrett/m/mfms/notes_c/irreg_sing_pt.pdf, and http://www.math.umn.edu/~garrett/m/mfms/notes_c/frobenius_ode.pdf ... In particular, this is not about PDEs, but about the ODEs obtained after various separations of variables.)

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From the point of view of analytic number theory, the point usually would be asymptotic behaviour. This is typically well understood, and is in the massive book of Watson. Apart from that, yes, numerous identities (which are not that deep, I feel), and an analogy with Kloosterman sums, though some would read that the other way round. A part of special function theory that appears quite well explored.

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For an analytic number theorist the most useful way to look at Bessel functions and related special functions (e.g. Whittaker functions or products of two such functions) is through their Mellin transforms. The Mellin transform explains:

(1) the shape of gamma factors in various automorphic $L$-functions

(2) the shape of Voronoi summation (as in your example)

(3) the behavior of the Bessel function at zero and at infinity

(4) the Taylor expansion of the Bessel function

The differential equation is indispensable when one needs to estimate the Hankel-type transforms arising in (2), or the Bessel transforms that arise in the Petersson and Kuznetsov formulae. Techniques involve integration by parts or passing to the Mellin side and deforming the contour.

Studying the formulae in Gradshteyn-Ryzhik is very useful for an analytic number theorist, and here I mean the whole book, not just the parts with Bessel functions!

For a conceptual understanding I recommend Part II of Representation theory and noncommutative harmonic analysis 2 (Enc.Math.Sci.59, Springer): Representations of Lie groups and special functions by Klimyk and Vilenkin.

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Expanding on Qiaochu's and David's comments, from the point of view of automorphic forms and automorphic representations, a good reference is "Special Functions and the Theory of Group Representations" by N. Ja. Vilenkin and published by the AMS.

"Special Functions" by Andrews, Askey, and Roy published by CUP is another good reference with an anayltic number theory slant.

EDIT: The point here is that for a finite group $G$ and a representation $\pi$ into a finite dimensional vector space $V$ with orthonormal basis $\{\vec{e_i}\}$, the function $g\to \langle \vec{e_i},\pi(g)\vec{e_j}\rangle$ gives the $i,j$ coefficient of the matrix associated to $\pi(g)$, or 'matrix coefficient function'. For a Lie group $G$ with an infinite dimensional representation $\pi$ in a Hilbert space $H$, there is a natural analog, and the functions which arise this way play an obviously important role in harmonic analysis in $G$. Bessel functions can be interpreted this way.

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One way is to think of Bessel functions as some sort of non-abelian analogues of the exponential function. In fact, you can form a complete orthogonal system of $L^2 (\mathbb{R^+}, x^{-1}d x)$ using the Bessel functions. See Chapter 16 of Iwaniec-Kowalski (page 411).

For manipulations, the formulae given in Appendix B of Iwaniec's Spectral Methods book usually suffice.

Yes, Dan Brown is more appropriate than Gradhsteyn and Rizhik for airport reading.

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I certainly can't claim much knowledge on these questions and the following remark is elementary, but Apostol wonderful exposition (in ``Modular Functions and Dirichlet Series in Number Theory", Springer GTM number 41, pages 94-110, second edition) of Rademacher's formula for the partition function $p(n)$ shows very clearly how the "Hardy-Littlewood-Ramanujan Circle Method" gives (page 109) : $$ p(n)=2\pi\big(\dfrac{\pi}{12}\big)^{3/2}\sum\limits_{k=1}^{\infty}A_k(n)k^{-5/2}\dfrac{1}{2i\pi}\int_{c-i\infty}^{c+i\infty}t^{-5/2} \exp \Big\{ t+\dfrac{\pi^2}{6k^2}\big(n-\dfrac{1}{24}\big)\dfrac{1}{t} \Big\} dt $$ where $~c=\pi/12,~~~s(h,k)~~$ is a Dedekind sum and $~~A_k(n)=\sum\limits_{0\leq h<k,gcd(h,k)=1} \exp(i\pi s(h,k)-2i\pi nh/k)$ .\ Now, in this (impressive...) formula, the integral is (almost) the Bessel function $~I_{3/2}~$ in thin disguise.

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