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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.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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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 $z\rightarrow y^2$ 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 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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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 $z\rightarrow y^2$ 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 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.)