0,2

Analogous to the infinitesimal Pascal matrix (m=0), A132440.

In general the matrix T begins (here m=1)

0;

m+1,0;

0, m+2, 0;

0, 0, m+3, 0;

0, 0, 0, m+4, 0;

Let LM(t) = exp(t*T) = limit [1 + t*T/n]^n as n tends to infinity.

Laguerre matrix(m) = [bin(n+m,k+m)] = LM(1) = exp(T) = [ revert of A074909 for m=1 ]. 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 Lag matrix(m) = LM(-1) = exp(-T)

Umbrally LM[b(.)] = exp(b(.)*T) = [ bin(n+m,k+m) * b(n-k) ]

A(j) = T^j / j! equals the matrix [bin(n+m,k+m) * 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 Laguerre(m) matrix. Hence the A(j)'s form a linearly independent basis for all matrices of the form [bin(n+m,k+m) d(n-k)].

For sequences with b(0) = 1, umbrally,

LM[b(.)] = exp(b(.)*T) = [ bin(n+m,k+m)] * b(n-k) ] .

[LM[b(.)]]^(-1) = exp(c(.)*T) = [ bin(n+m,k+m)] * 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+m) * a(n-1),

2) B(x) = x^(-m) (xDx) x^m A(x)

3) B(x) = x * Lag(1,-:xD:,m) A(x) = x * [(m+1) + xD] A(x)

4) EB(x) = D^(m) * (x) * D^(-m) EA(x) where D is the derivative w.r.t. x, (:xD:)^j = x^j*D^j, Lag(n,x,m) is the associated Laguerre polynomial and D^(-m) x^n / n! = x^(m+n) / (m+n)! are Riemann-Liouville integrals.

So the exponentiated operator can be characterized (with loose notation) as

5) exp(t*T) A(x) = x^(-m) exp(t*xDx) x^m A(x) = [sum(n=0,1,...) (t*x)^n * Lag(n,-:xD:m)] A(x) = [exp{[t*u/(1-t*u)]*:xD:} / (1-t*u)^(m+1) ] A(x) (eval. at u=x) = A[x/(1-t*x)]/(1-t*x)^(m+1), a generalized Euler transformation for an o.g.f.,

6) exp(t*T) EA(x) = D^(m) exp(t*x) D^(-m) EA(x) = [D/(D-1)]^m exp[(t+a(.))*x] = exp(t*x) [(t+D)/D]^m EA(x)

7) exp(t*T) * a = LM(t) * a = [sum(k=0,...,n) bin(n+m,k+m)* t^(n-k) * a(k)], a vector array.

With t=1 and a(k) = (-x)^k / k!, then LM(1) * a = [Laguerre(n,x,m)], a vector array with index n and the o.g.f. A(x) becomes transformed into the e.g.f. for the associated Laguerre polynomials of order m.

The exponential formulas can be umbrally extended as in A132440. And, the formulas can be extended to non-integer m.

Table of n, a(n) for n=0..77.

T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras

G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009.

M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3

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[(m+1)+ RL] 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

Table[PadLeft[{n, 0}, n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 30 2014 *)

Sequence in context: A091731 A284269 A140579 * A127648 A212209 A259481

Adjacent sequences: A132678 A132679 A132680 * A132682 A132683 A132684

easy,nonn,tabl

Tom Copeland, Nov 15 2007, Nov 16 2007, Nov 27 2007

Missing 0 added to array by Tom Copeland, Aug 02 2013

approved