using SpecialFunctions
include("helper.jl")

# Gaussian potentials in HO space
inv_factorial(n) = Iterators.prod(inv.(1:n))
sqrt_factorial(n) = Iterators.prod(sqrt.(n:-1:1))
N_lnk(l, n, k) = 1/sqrt_factorial(n) * binomial(n, k) * sqrt(gamma(n + l + 3/2)) / gamma(k + l + 3/2)
Talmi(l, R, k1, k2; μω_gen=1.0) = (-1)^(k1 + k2) * (1 + 1/(μω_gen * R^2))^-(3/2 + l + k1 + k2) * gamma(3/2 + l + k1 + k2)
V_Gaussian(R, l, n1, n2; μω_gen=1.0) = (-1)^(n1 + n2) * better_sum([N_lnk(l, n1, k1) * N_lnk(l, n2, k2) * Talmi(l, R, k1, k2; μω_gen=μω_gen) for (k1, k2) in Iterators.product(0:n1, 0:n2)])

# for numerical evaluation of V matrix elements
sqrt_double_factorial(n) = Iterators.prod(sqrt.(n:-2:1))
sqrt_sqrt_pi = sqrt(sqrt(pi))
laguerre(l, n, x) = gamma(n + l + 3/2) * better_sum([(-x * x)^k / gamma(k + l + 3/2) * inv_factorial(n - k) * inv_factorial(k)  for k in 0:n])
ho_basis(l, n, x) = (-1)^n / sqrt_sqrt_pi * 2^((n + l + 2) / 2) * sqrt_factorial(n) / sqrt_double_factorial(2*n + 2*l + 1) * x^(l + 1) * exp(-x^2 / 2) * laguerre(l, n, x)