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

%I A218272 #45 Nov 22 2023 04:26:54
%S A218272 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,
%T A218272 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,
%U A218272 0,0,0,0,0,0,0,0,11,0,0,0
%N A218272 Infinitesimal generator for transpose of the Pascal matrix A007318 (as upper triangular matrices).
%C A218272 T is the transpose of A132440.
%C A218272 Let M(t) = exp(t*T) = limit [1 + t*T/n]^n as n tends to infinity.
%C A218272 Then M(1) = the transpose of the lower triangular Pascal matrix A007318, with inverse M(-1).
%C A218272 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
%C A218272   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).
%C A218272 See A132440 as an analog and more general discussion.
%C A218272 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
%C A218272 From _Tom Copeland_, Jul 02 2018: (Start)
%C A218272 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.
%C A218272 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.
%C A218272 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,...).
%C A218272 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).
%C A218272 (End)
%H A218272 Tom Copeland, <a href="http://tcjpn.wordpress.com/2012/11/29/infinigens-the-pascal-pyramid-and-the-witt-and-virasoro-algebras/">Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras</a>
%H A218272 P. Olver, <a href="http://www.math.umn.edu/~olver/di_/contact.pdf">The canonical contact form</a>.
%F A218272 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):
%F A218272   1) b(n) = (n+1) * a(n+1),
%F A218272   2) B(x) = D A(x), or
%F A218272   3) EB(x) = DxD  EA(x),
%F A218272   where D is the derivative w.r.t. x.
%F A218272 So the exponentiated operator can be characterized as
%F A218272   4) exp(t*T) A(x) = exp(t*D) A(x) = A(x+t),
%F A218272   5) exp(t*T) EA(x) = exp(t*DxD) EA(x) = exp[x*a/(1+t*a)]/(1+t*a),
%F A218272      = Sum_{n>=0} (1+t*a)^(-n-1) (x*a)^n/n!, where umbrally
%F A218272      a^n *(1+t*a)^(-n-1) = Sum_{j>0} binomial(n+j,j)a(n+j)t^j,
%F A218272   6) exp(t*T) EA(x) = Sum_{n>=0} a(n) t^n Lag(n,-x/t),
%F A218272      where Lag(n,x) are the Laguerre polynomials (A021009), or
%F A218272   7) [exp(t*T) * a]_n = [M(t) * a]_n
%F A218272      = Sum_{j>=0} binomial(n+j,j)a(n+j)t^j.
%F A218272 For more on the operator DxD, see A021009 and references in A132440.
%e A218272 Matrix T begins
%e A218272   0,1;
%e A218272   0,0,2;
%e A218272   0,0,0,3;
%e A218272   0,0,0,0,4;
%e A218272   0,0,0,0,0,5;
%e A218272   0,0,0,0,0,0,6;
%e A218272   ...
%t A218272 Table[PadLeft[{n+1}, n+2], {n, 0, 11}] // Flatten (* _Jean-François Alcover_, Apr 30 2014 *)
%Y A218272 Essentially the same as A134402, A132440 and A130460.
%K A218272 nonn,easy,tabf
%O A218272 0,5
%A A218272 _Tom Copeland_, Oct 24 2012