For any polynomial function f(x) if you know f(a), f'(a), f"(a), f"'(a)... for some value of a you can reconstruct the function. This is true even if the polynomial has an infinite number of terms.

(f' is the first derivative of f, f" is the second derivative etc.)

Define s(a) to be the sequence of f(a), f'(a), f"(a), f"'(a), ...

For each of the following, find the function:

(1) s(1) = 19, 23, 32, 18, 0, 0, 0, 0, ...

(2) s(0) = 1, 2, 4, 8, 16, 32, ...

(3) s(1) = 1, -1, 1/2, -1/6, 1/24, -1/120, 1/720, ...

(4) s(0) = 0, 1, 0, -1, 0, 1, 0, -1, ...

(5) s(0) = ln(2), ln(2^ln(2)), ln(2^{ln(2ln(2) )}), ...

(In reply to Partially done but help needed (Spoiler) by Harry)

Great analysis, Harry!
Sorry, I don't have a closed form for that pesky (3) - but I doubt it's a "Bessel" function ...    ;-)

Posted by JayDeeKay on 2009-03-30 13:30:09