A218234 Infinitesimal generator for padded Pascal matrix A097805 (as lower triangular matrices).

0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0

Offset: 0

Author

Tom Copeland, Oct 24 2012

Keywords

• easy
• tabl
• nonn

Comments

Matrix T begins

0;

0,0;

0,1,0;

0,0,2,0;

0,0,0,3,0;

0,0,0,0,4,0;

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

Then M(1) = the lower triangular padded Pascal matrix A097805, with inverse M(-1).

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^2*L 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).

See A132440 for an analog and more general discussion.

Links

• P. Blasiak and P. Flajolet, Combinatorial models of creation-annihilation
• T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
• T. Copeland, Mathemagical Forests
• T. Copeland, Addendum to Mathemagical Forests
• G. Dattoli, B. Germano, M. Martinelli, and P. Ricci, Touchard like polynomials and generalized Stirling polynomials
• W. Lang, Combinatorial interpretation of generalized Stirling numbers

Programs

Mathematica

Table[PadLeft[{n-1, 0}, n+1], {n, 0, 12}]  // Flatten (* Jean-François Alcover, Apr 30 2014 *)

Formula

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(1) = 0, b(n) = (n-1) * a(n-1),

2) B(x) = x^2D A(x)= x (xDx)(1/x)A(x) = x^2 * Lag(1,-:xD:) A(x)/x , or

3) EB(x) = D^(-1)xD EA(x),

where D is the derivative w.r.t. x, (D^(-1)x^j/j!) = x^(j+1)/(j+1)!, (:xD:)^j = x^j*D^j, and Lag(n,x) are the Laguerre polynomials A021009.

So the exponentiated operator can be characterized as

4) exp(t*T) A(x) = exp(t*x^2D) A(x) = x exp(t*xDx)(1/x)A(x)

= x [sum(n=0,1,...) (t*x)^n * Lag(n,-:xD:)] A(x)/x

= x [exp{[t*u/(1-t*u)]*:xD:} / (1-t*u) ] A(x)/x (eval. at u=x)

= A[x/(1-t*x)], a special Moebius or linear fractional trf.,

5) exp(t*T) EA(x) = D^(-1) exp(t*x)D EA(x), a shifted Euler trf.

for an e.g.f., or

6) [exp(t*T) * a]_n = [M(t) * a]_n

= [sum(k=0,...,n-1) binomial(n-1,k)* t^(n-1-k) * a(k+1)] with [M(t) * a]_0 = a_0

For generalizations and more on the operator x^2D, see A132440 and the references therein and above, and A094638.