OFFSET
0,2
COMMENTS
The matrix T begins
0;
2, 0;
0, 6, 0;
0, 0, 12 0;
0, 0, 0, 20, 0;
Along the nonvanishing diagonal the n-th term is (n+2)*(n+1).
Let LM(t) = exp(t*T) = lim_{n->infinity} (1 + t*T/n)^n.
Shifted Lah matrix = [bin(n+1,k+1)*(n)!/(k)! ] = LM(1) = exp(T). Truncating the series gives the n X n submatrices. In fact, the submatrices of T are nilpotent with [Tsub_n]^(n+1) = 0 for n=0,1,2,....
Inverse shifted Lah matrix = LM(-1) = exp(-T)
Umbrally shifted Lah[b(.)] = exp(b(.)*T) = [ binomial(n+1,k+1)*(n)!/(k)! * b(n-k) ]
A(j) = T^j / j! equals the matrix [binomial(n+1,k+1)*(n)!/(k)! * delta(n-k-j)] where delta(n) = 1 if n=0 and vanishes otherwise (Kronecker delta); i.e. A(j) is a matrix with all the terms 0 except for the j-th lower (or main for j=0) diagonal which equals that of the Lah matrix. Hence the A(j)'s form a linearly independent basis for all matrices of the form [binomial(n+1,k+1) * (n)! / (k)! * d(n-k)].
For sequences with b(0) = 1, umbrally,
LM[b(.)] = exp(b(.)*T) = [ bin(n+1,k+1)*(n)!/(k)! * b(n-k) ] .
[LM[b(.)]]^(-1) = exp(c(.)*T) = [ bin(n+1,k+1)*(n)!/(k)! * c(n-k) ] where c = LPT(b) with LPT the list partition transform of A133314. Or,
[LM[b(.)]]^(-1) = exp[LPT(b(.))*T] = LPT[LM(b(.))] = LM[LPT(b(.))] = LM[c(.)] .
The matrix operation b = T*a can be characterized in several ways in terms of the coefficients a(n) and b(n), their o.g.f.'s A(x) and B(x), or e.g.f.'s EA(x) and EB(x).
1) b(0) = 0, b(n) = (n+1)*(n) * a(n-1),
2) B(x) = x * D^2 * x^2 A(x)
3) B(x) = x * 2 *Lag(2,-:xD:,0) A(x)
4) EB(x) = D * x^2 EA(x)
where D is the derivative w.r.t. x, (:xD:)^j = x^j*D^j and Lag(n,x,m) is the associated Laguerre polynomial of order m.
The exponentiated operator can be characterized (with loose notation) as
5) exp(t*T) * a = LM(t) * a = [sum(k=0,...,n) bin(n+1,k+1) * n!/k! t^(n-k) * a(k)] = [ t^n * n! * Lag(n,-a(.)/t,1) ], a vector array.
With t=1 and a(k) = (-x)^k, then LM(1) * a = [ n! * Laguerre(n,x,1) ], a vector array with index n .
6) exp(t*T) EA(x) = EB(x) = EA[ x / (1-x*t) ] / (1-x*t)^2
LINKS
G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009, p. 4
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
FORMULA
Given a polynomial sequence p_n(x) with p_0(x)=1 and the lowering and raising operators L and R defined by L P_n(x) = n * P_(n-1)(x) and R P_n(x) = P_(n+1)(x), the matrix T represents the action of R*L^2*R^2 in the p_n(x) basis. For p_n(x) = x^n, L = D = d/dx and R = x. For p_n(x) = x^n/n!, L = DxD and R = D^(-1). - Tom Copeland, Oct 25 2012
MATHEMATICA
Table[PadLeft[{n*(n-1), 0}, n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 30 2014 *)
CROSSREFS
KEYWORD
AUTHOR
Tom Copeland, Nov 15 2007, Nov 16 2007, Nov 27 2007
STATUS
approved