Binomial Coefficient Identity

[mj]

i am assuming that f is defined by a power series and that the operator norm of A is less than the radius of convergence of the power series.



the definition of f(A) is then clear.



ditto for any for any other ops we are considering



define R to be the subset of polynomials h such that h(f(A),g(A)) has the some coefficients in its power series for matrices , as it does for real variables, h(f(x),g(x))



clearly, R contains 1 x and y



R is clearly closed under addition and multiplication (just check the coefficients are defined by the same operations)



so R is all polynomials.

[AVt]

Äääh Blush thx ... ok, then it's just re-arranging terms in series ... beautiful simple!