Hermite Transform / Inverse Hermite Transform

Function computes finite Hermite transform or its inverse transform.

Forward: Function input c is the ordinary power series coefficient vector. Function output fh is the Hermite transform coefficient vector.
fh = hermite_transform(c)

Inverse: Function output c is the ordinary power series coefficient vector. Function input fh is the Hermite transform coefficient vector.
c = inverse_hermite_transform(fh)

Thorough theory can be found here
https://qr.ae/pNrYxL

Quick explanation
Define an ordinary power series, where c_k | k=0,1,2... denotes the ordinary coefficient vector
y(x) = c_0 + xc_1 + x^2c_2...
Define a Hermite transform , where fh_k | k=0,1,2... denotes the Hermite coefficient vector in terms of Hermite polynomials
y(x) = r_0 + H_1(x)r_1+H_2(x)r_2...
fh_k = sqrt(pi)(2^k)k!r_k

Forward transform:
Function input is the finite ordinary coefficient vector c_k | k=0,1,2...K and function output is equal-length Hermite coefficient vector fh_k | k=0,1,2...K

Inverse transform:
Function output is the finite ordinary coefficient vector c_k | k=0,1,2...K and function input is equal-length Hermite coefficient vector fh_k | k=0,1,2...K
Inverse

Example: Calculate fh_k given the Taylor polynomial coefficients c_k estimating the power series expansion of y(x) = e^x about x=0, then invert the transformation to restore c

Solution:
c = 1./factorial(0:15); %exponential taylor polynomial degree 15
fh = hermite_transform(c); %calculate fh
cr = inverse_hermite_transform %restore c exactly, call cr

Thorough theory can be found here
https://qr.ae/pNrYxL