I am asked:
For any real number x and positive integer k, define the notation [x,k] by the recursion [x,k+1] = (x-k) [x,k] and [x,1] = x.
If n is any positive integer, one can now express the monomial x^n as a polynomial in [x,1], [x,2], . . . , [x,n]. Find a general formula that accomplishes this, and prove that your formula is correct.
I am struggling finding the final formula.
I multiplied some of the polynomials out to get
$$x=[x,1]$$ $$x^2=[x,2]+[x,1]$$ $$x^3=[x,3]+3[x,2]+2[x,1]$$ $$x^4=[x,4]+6[x,3]+3[x,2]+4[x,1]$$
I am stumped where to go from here. I am unsure of how to create a formula from this, I feel like the trailing coefficients would be a factorial but am not sure if this is helpful in this scope.