A fairly direct consequence of the indefinite sum m Sum Pochhammer(k,p) = Pochhammer(m,p+1)/(p+1), k=1
which is in turn a direct consequence of the rule of formation of Pascal's triangle,
nCk + nC(k+1) = (n+1)C(k+1), is
bern(p) = -(-1)^j*(p+j)*sum((-1)^k*stirling_s1(k+1,j)*stirling_s2(p+j-1,k)/(k+1),k,0,p+j-1)/binom(p+j,j) p + j - 1 k ==== (- 1) s S j \ k + 1, j p + j - 1, k (- 1) (p + j) > ------------------------------ / k + 1 ==== k = 0 B = - -------------------------------------------------------, p>0. p binom(p + j, j)
For p:=1,
bern(p) = sum((-1)^k*k!*stirling_s2(p,k)/(k+1),k,0,p)
p k ==== (- 1) k! S \ p, k B = > ---------------. p / k + 1 ==== k = 0
This latter, at least, must be in G, Knuth, and P. --rwg
Much better: eqn (38) of http://mathworld.wolfram.com/BernoulliPolynomial.html is the generalization of the latter to Bernoulli polynomials.
Ah, but it's equivalent (or very nearly) to the former if we replace p by n-j, multiply by x^j, and sum over j. This got me Faulhabers instead of Bernpolys: bernpoly(x,n)-bern(n) = n*'sum((-1)^k*stirling_s2(n-1,k-1)*pochhammer(-x,k)/k,k,1,n) n k ==== (- 1) S (- x) \ n - 1, k - 1 k B (x) - B = n > --------------------------- n n / k ==== k = 1 but they can't be this close and inequivalent. In a later msg, I said
Similar identities, with any desired degree of sparseness can be had from multisecting the standard identity ...
Here is an "every 12th" bernpoly recurrence: 12*sum(binom(n,12*k)*x^(12*k)*bernpoly(y,n-12*k),k,0,floor(n/12)) = bernpoly(y+%i*x,n)+bernpoly(y-%i*x,n)+bernpoly(y+%i^(1/3)*x,n)+bernpoly(y-%i^(1/3)*x,n)+bernpoly(y+x/%i^(1/3),n)+bernpoly(y-x/%i^(1/3),n)+bernpoly(y+(-1)^(2/3)*x,n)+bernpoly(y-(-1)^(2/3)*x,n)+bernpoly(y+(-1)^(1/3)*x,n)+bernpoly(y-(-1)^(1/3)*x,n)+bernpoly(y+x,n)+bernpoly(y-x,n) n floor(--) 12 ==== \ 12 k 12 > binomial(n, 12 k) x B (y) = / n - 12 k ==== k = 0 1/3 1/3 x B (y + %i x) + B (y - %i x) + B (y + %i x) + B (y - %i x) + B (y + -----) n n n n n 1/3 %i x 2/3 2/3 1/3 + B (y - -----) + B (y + (- 1) x) + B (y - (- 1) x) + B (y + (- 1) x) n 1/3 n n n %i 1/3 + B (y - (- 1) x) + B (y + x) + B (y - x). n n n --rwg CR*P! I meant between C and Aleph_1 ! Why is my proofreader in Albuquerque?