Add support for Laguerre

[dlfivefifty]

No description provided.

[MikaelSlevinsky]

I guess the Hilbert transform would require a good implementation of the exponential integral.

[MikaelSlevinsky]

(assuming we're working with exponentially weighted Laguerre polynomials)

[dlfivefifty]

Something like H[L_k^1/2 sqrt(x)exp(-x)] = L_k^-1/2 should be true.

Cauchy transform would just follow from Olver's algorithm and first moment.

[MikaelSlevinsky]

Ah, weighted generalized Laguerre! It's less complicated than H[L_k exp(-x)]

[dlfivefifty]

👍

[dlfivefifty]

H[L_k exp(-x)] has the same annoying logarithmic singularities as H[P_k] that make it not a nice basis. Note that any other Jacobi singularities do not have this annoying feature.

[MikaelSlevinsky]

I think that H[P_k^{(alpha,beta)}] has a lurking log singularity if alpha and beta are both integer, since there's a banded conversion to P_k^{(0,0)}. But for positive alpha and beta, the singularity would be weaker than the Legendre case.

[dlfivefifty]

Good point

[MikaelSlevinsky]

It turns out that by Theorem 4.62.1 in Szegő's Orthogonal Polynomials, my edition, there should be (weak) log singularities at the respective endpoint corresponding to integer alpha or beta (not just when they're both integer).