(50g) Bernoulli polynomials

[ttw]

There are some direct formulas without integrations (although I always use integration because it's non-numerical and easier to remember.)

B_m(X) = Sum(n:0:m)[1/(n+1)Sum(k:0:n)[(-1)^k*C(n,k)*(x+k)^m]

It's easier to start with 1 and integrate then scale the leading coefficient to 1 and add the constant to get the average to be zero. That's probably equivalent to what you are doing. It can all be done symbolically.

It's useful for periodizing functions before doing some types of integrations. The results are much "smoother" (fewer high frequency left over terms) than most periodization methods.