Special functions

[blefloch]

• See paper that M.R. gave me: describe zeta-function regularization, its representation as exp(partial(zeta)@0), the integral representation of zeta in terms of sums of exponentials, and issues of signs.
• Introduce Gamma and multiple Gammas as zeta-regularized products. Each time describe in physics terms if possible (e.g. "Gamma is the one-loop det of a 2d (2,2) chiral")
• Plain old \Gamma, its friend little \gamma, Pochhammer, hypergeometric function
• Jacobi \theta functions
• zeta, Hurwitz zeta
• \Gamma_b, S_b (one-loop det in 3d N=2), \Upsilon_b (1/one-loop det of 4d N=2 hyper)
• elliptic Gamma, (p;p)_\infty
• elliptic hypergeometric
• Spiridonov-type integrals?

Avoid the more complicated properties, just keep the minimum to remember what these functions are. Check license of the http://dlmf.nist.gov/ library of maths functions.

[blefloch]

Families of polynomials: each of these is a polynomial of degree n in the variable x.

• Legendre polynomials are special Gegenbauer polynomials P_n(x)=C_n^{(1/2)}(x).
• Gegenbauer polynomials are special Jacobi polynomials C_n^{(\alpha+1/2)}(x)=\frac{(2\alpha+1)_n}{(\alpha+1)_n} P_n^{(\alpha,\alpha)}(x). Recursion nC_n^{(\gamma)}=2x(n-1+\gamma)C_{n-1}^{(\gamma)}-(n-2+2\gamma)C_{n-2}^{(\gamma)} with C_0^{(\gamma)}=1 and C_1^{(\gamma)}=2\gamma x. Used in expansion 1/|x-y|^{2\gamma}=\sum_{n=0}^{\infty} C_n^{(\gamma)}(\cos\theta) |y|^k/|x|^{k+1} hence in conformal blocks.
• Jacobi polynomials P_n^{(\alpha,\beta)}(x)=\frac{(\alpha+1)_n}{n!} {_2F_1}(-n,1+\alpha+\beta+n;1+\alpha;(1-x)/2). Orthogonal wrt weight w(x)=(1-x)^\alpha (1+x)^\beta on [-1,1]. Rodriguez formula w(x)P_n(x)=\frac{(-1)^n}{n! 2^n} \partial_x^n [w(x) (1-x^2)^n]. Differential equation (1-x^2)y''+(\beta-\alpha-(\alpha+\beta+2)x)y'+n(n+\alpha+\beta+1)y=0.