G(x1+x2)=\sum g(a_{i,j}x^ix^j)

[John Creighto]

I wonder in what circumstances:

given a function g, we can analytically find the coefficients $$a_{i,j}$$

$$g(x1+x2)=\sum a_{i,j}g(x^ix^j)$$

I'm not if this would server any useful applications but the transformation looks interesting to me. It looks very simmilar to a polynomial transformation.

 

[flatmaster]

I wonder in what circumstances:

given a function g, we can analytically find the coefficients $$a_{i,j}$$

$$g(x1+x2)=\sum a_{i,j}g(x^ix^j)$$

I'm not if this would server any useful applications but the transformation looks interesting to me. It looks very simmilar to a polynomial transformation.

For an arbitrary function g(x), I doubt it. There might be a family of functions though.