1,5

The lowering (or delta) operator for these polynomials is L = (D-1)/2 + T{ (1/2) * exp[(D-1)/2] } and the raising operator is R = 2t * { 1 - T[ (1/2) * exp[(D-1)/2] ] }, where T(x) is the tree function of A000169. In addition, L = E(D,1) = A(D) where E(x,t) is the e.g.f. of A134991 and A(x) is the e.g.f. of A000311, so L = sum(j=1,...) A000311(j) * D^j / j! also. The polynomials and operators can be generalized through A134991.

S. Roman, The Umbral Calculus, Academic Press, New York, 1984.

G. Rota, Finite Operator Calculus, Academic Press, New York, 1975.

Row polynomials are P(n,t) = sum(j=1,...,n) C(n,j) * x^j = [ Bell(.,-t) + 2t ]^n,umbrally, where Bell(j,t) are the Touchard/Bell/exponential polynomials described in A008277, with P(0,t) = 1 .

The e.g.f. is exp{ t * [ -exp(x) + 2x + 1] } and [ P(.,t) + P(.,s) ]^n = P(n,s+t) .

The lowering operator gives L[P(n,t)] = n * P(n-1,t) = (D-1)/2 * P(n,t) + sum(j=1,...) j^(j-1) * 2^(-j) / j! * exp(-j/2) * P(n,t + j/2) .

The raising operator gives R[P(n,t)] = P(n+1,t) = 2t * { P(n,t) - sum(j=1,...) j^(j-1) * 2^(-j) / j! * exp(-j/2) * P(n,t + j/2) } .

Therefore P(n+1,t) = 2t * { [ (1+D)/2 * P(n,t) ] - n * P(n-1,t) } .

P(n,1) = (-1)^n * A074051(n) and P(n,-1) = A126617(n) .

See Rota, Roman, Mathworld or Wikipedia on Sheffer sequences and umbral calculus for more formulae, including expansion theorems.

Sequence in context: A046643 A112475 A126799 this_sequence A016566 A096744 A160752

Adjacent sequences: A135491 A135492 A135493 this_sequence A135495 A135496 A135497

sign,tabl

Tom Copeland (tcjpn(AT)msn.com), Feb 08 2008