Let

    G:=s->q(1-ps)^(-1)

be a generating function with 1-p=q in (0,1). Then the repeated function composition

    G(G(G(s)));(G@@3)(s);simplify(%);

gives me some concrete expression. What is the simplest way to get the general expression for 3 replaced by n?   

For m=1,2, how do I show with Maple that the first two moments of the Borel-Tanner distribution are simple functions of k and lamda, e.g., k/(1-lambda) for the mean? How do I get the closed-form expressions with maple? Code:

simplify(sum(x^m*k*x^(x-k-1)*lambda^(x-k)*exp(-lambda*x)/factorial(x-k), x = k .. infinity)) assuming lambda > 0, lambda < 1, k::posint; evalf(subs(k = 1, lambda = .8, %))

Consider the identity 4*cos(theta)^3-3*cos(theta) = cos(3*theta). Does maple know about it, or how to prove the identity, which is true by deMoivre's formula, with maple? The trig option for simplify seems not enough. Thanks.

There is some advice how to solve a cubic trigonometrically  here, but nothing presentable seems to appear out of it. Could someone please demonstrate how to do that with maple for a general (monic) cubic. Thanks.

Why does the equality fail to hold in case of a positive geometric density (with the location parameter v=1,2,... if in general)? What's wrong: symbolical result or numerical evaluation? Thanks.
with(Statistics):
X := RandomVariable(Geometric(p))+1; M := Mean(X); A := simplify((X-M)/M, size);
Mean(1-A+A^2/(1+A))/M = (Mean(1)-Mean(A)+Mean(A^2/(1+A)))/M; is(%); subs(p = 1/3, `%%`); simplify(%, size); evalf(%)