A218272 Infinitesimal generator for transpose of the Pascal matrix A007318 (as upper triangular matrices).

0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0

Offset: 0

Author

Tom Copeland, Oct 24 2012

Keywords

• nonn
• easy
• tabf

Comments

T is the transpose of A132440.

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

Then M(1) = the transpose of the lower triangular Pascal matrix A007318, 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 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 as an analog and more general discussion.

Sum_{n>=0} c_n T^n / n! = e^(c.T) gives the Maurer-Cartan form matrix for the one-dimensional Leibniz group defined by multiplication of a Taylor series by the formal Taylor series e^(c.x) (cf. Olver). - Tom Copeland, Nov 05 2015

From Tom Copeland, Jul 02 2018: (Start)

The transpose Psc^Trn of the lower triangular Pascal matrix Psc = A007318 gives the numerical coefficients of the Maurer-Cartan form matrix M of the Leibniz group Leibniz(n)(1,1) presented on p. 9 of the Olver paper. M = exp[c. * T] with (c.)^n = c_n and T the Lie infinitesimal generator of this entry. The columns e^T are the rows of the Pascal matrix A007318.

M can be obtained by multiplying each n-th column vector of Psc by c_n and then transposing the result; i.e., with the diagonal matrix H = Diag(c_0, c_1, c_2, ...), M = (Psc * H)^Trn = H * Psc^Trn.

M is a matrix representation of the differential operator S = e^{c.*D} with D = d/dx, which acting on x^m gives the Appell polynomial p_m(x) = (c. + x)^m, with (c.)^k = c_k an arbitrary indeterminate except for c_0 = 1. For example, S x^2 = (c. + x)^2 = c_0*x^2 + 2*c_1*x + c_2, and M * (0,0,1,0,0,...)^Trn = (c_2,2*c_1,c_0,0,0,...)^Trn = V, so V^Trn = (0,0,1,0,...) * M^Trn = (0,0,1,0,...) * Psc * H = (c_2,2*c_1,c_0,0,...).

The differential lowering and raising operators for the Appell sequence are given by L = D and R = x + dlog(S)/dD, with L p_n(x = n * p_(n-1)(x) and R p_n(x) = p_(n+1)(x).

(End)

Examples

			Matrix T begins
  0,1;
  0,0,2;
  0,0,0,3;
  0,0,0,0,4;
  0,0,0,0,0,5;
  0,0,0,0,0,0,6;
  ...
		

Links

• Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
• P. Olver, The canonical contact form.

Crossrefs

Essentially the same as A134402, A132440 and A130460.

Programs

Mathematica

Table[PadLeft[{n+1}, n+2], {n, 0, 11}] // 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(n) = (n+1) * a(n+1),

2) B(x) = D A(x), or

3) EB(x) = DxD EA(x),

where D is the derivative w.r.t. x.

So the exponentiated operator can be characterized as

4) exp(t*T) A(x) = exp(t*D) A(x) = A(x+t),

5) exp(t*T) EA(x) = exp(t*DxD) EA(x) = exp[x*a/(1+t*a)]/(1+t*a),

= Sum_{n>=0} (1+t*a)^(-n-1) (x*a)^n/n!, where umbrally

a^n *(1+t*a)^(-n-1) = Sum_{j>0} binomial(n+j,j)a(n+j)t^j,

6) exp(t*T) EA(x) = Sum_{n>=0} a(n) t^n Lag(n,-x/t),

where Lag(n,x) are the Laguerre polynomials (A021009), or

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

= Sum_{j>=0} binomial(n+j,j)a(n+j)t^j.

For more on the operator DxD, see A021009 and references in A132440.