A132681 Infinitesimal generator matrix for a diagonally-shifted Pascal matrix, binomial(n+m,k+m), for m=1, related to Laguerre(n,x,m).

0, 2, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0

Offset: 0

Author

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

Keywords

• easy
• nonn
• tabl

Comments

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.

Links

• 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

Programs

Mathematica

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

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[(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

Extensions

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