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The lie groups + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's equation.

The standard references are Miller & Vilenkin, but they are far too dense, too much unnecessary information for my purposes & I have no idea/time how to sift through it all. I'd really like a short, direct exposition of this material as it relates to the special functions you get from separating the Laplacian like the Hypergeometric, Legendre, Bessel, Airy etc... equations in a geometric way, if such a thing exists. I believe it's something like a geometric interpretation of Weisner's method.

For example, Bessel's equation seems to be saying, find a function in the plane such that when we shift it right, then shift it back left again, all locally (i.e. differentially), we get the same function:

(c.f. Killingbeck, Mathematical Technique's and Applications, sec. 8.21).

The idea is to take Bessel's equation, factor it, add an extra variable to make the factors parameter independent so that they become elements of a lie algebra, identify the meaning of those factors, in this case notice the Lie algebra factors are translations in Polar coordinates, and realize it's just a differential expression of a symmetry.

Bessel's equation arises from [itex]LRv = v[/itex] when you express [itex]L[/itex] & [itex]R[/itex] in polar coordinates. It makes sense to express them in polar coordinates since Bessel arises from separating the Laplacian assuming cylindrical symmetry, and the [itex]LRv = v[/itex] assumption (not [itex]LRv = w[/itex]) is motivated by the symmetry of the Laplacian.

Using this idea we can, for some reason, actually solve Bessel's equation with a picture!:

We just want to shift [itex]\mathcal{J}_n(r,\phi)[/itex] in the x-direction using the operator [itex]e^{a\tfrac{\partial}{\partial x}}[/itex] expressed in polar coordinates: [itex]e^{\tfrac{a}{2}(\mathcal{L}-\mathcal{R})}[/itex] and realize it will be equal to [itex]\mathcal{J}_n(r',\phi')[/itex]:

So the last line comes from dragging all this to the origin and putting it along the x-axis, here we see the geometric meaning of Bessel functions!

Hypergeometric is supposed to be related to [itex]SL(2,R)[/itex] symmetries, Bessel to translational planar symmetries, the Gamma function related to linear symmetries [itex]y = ax + b[/itex], etc... Is there an easy unified geometric exposition on how to deal with these babies?

References:

1. Killingbeck, Mathematical Technique's and Applications, sec. 8.21

2. Vilenkin, Representation of Lie Groups and Special Functions Vol. 1

3. Vilenkin, Special Functions and Theory of Group Representations

4. Miller, Lie Theory and Special Functions

5. Kaufman, Special Functions of Mathematical Physics from the Viewpoint of Lie

Algebra

*for the relationship between the special functions of mathematical physics, lie groups and representation theory? Something paralleling my example of Bessel's equation. Or maybe we could use this thread to post geometric explanations of all these ode's!***Would anyone have a non-dense/explicit/easy reference aimed at physicists**The standard references are Miller & Vilenkin, but they are far too dense, too much unnecessary information for my purposes & I have no idea/time how to sift through it all. I'd really like a short, direct exposition of this material as it relates to the special functions you get from separating the Laplacian like the Hypergeometric, Legendre, Bessel, Airy etc... equations in a geometric way, if such a thing exists. I believe it's something like a geometric interpretation of Weisner's method.

For example, Bessel's equation seems to be saying, find a function in the plane such that when we shift it right, then shift it back left again, all locally (i.e. differentially), we get the same function:

(c.f. Killingbeck, Mathematical Technique's and Applications, sec. 8.21).

The idea is to take Bessel's equation, factor it, add an extra variable to make the factors parameter independent so that they become elements of a lie algebra, identify the meaning of those factors, in this case notice the Lie algebra factors are translations in Polar coordinates, and realize it's just a differential expression of a symmetry.

Bessel's equation arises from [itex]LRv = v[/itex] when you express [itex]L[/itex] & [itex]R[/itex] in polar coordinates. It makes sense to express them in polar coordinates since Bessel arises from separating the Laplacian assuming cylindrical symmetry, and the [itex]LRv = v[/itex] assumption (not [itex]LRv = w[/itex]) is motivated by the symmetry of the Laplacian.

Using this idea we can, for some reason, actually solve Bessel's equation with a picture!:

We just want to shift [itex]\mathcal{J}_n(r,\phi)[/itex] in the x-direction using the operator [itex]e^{a\tfrac{\partial}{\partial x}}[/itex] expressed in polar coordinates: [itex]e^{\tfrac{a}{2}(\mathcal{L}-\mathcal{R})}[/itex] and realize it will be equal to [itex]\mathcal{J}_n(r',\phi')[/itex]:

So the last line comes from dragging all this to the origin and putting it along the x-axis, here we see the geometric meaning of Bessel functions!

Hypergeometric is supposed to be related to [itex]SL(2,R)[/itex] symmetries, Bessel to translational planar symmetries, the Gamma function related to linear symmetries [itex]y = ax + b[/itex], etc... Is there an easy unified geometric exposition on how to deal with these babies?

References:

1. Killingbeck, Mathematical Technique's and Applications, sec. 8.21

2. Vilenkin, Representation of Lie Groups and Special Functions Vol. 1

3. Vilenkin, Special Functions and Theory of Group Representations

4. Miller, Lie Theory and Special Functions

5. Kaufman, Special Functions of Mathematical Physics from the Viewpoint of Lie

Algebra

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