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%I #28 Apr 30 2014 06:40:34
%S 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,
%T 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,
%U 0,0,0,0,0,0,0,0,10,0,0,0
%N Infinitesimal generator for padded Pascal matrix A097805 (as lower triangular matrices).
%C Matrix T begins
%C 0;
%C 0,0;
%C 0,1,0;
%C 0,0,2,0;
%C 0,0,0,3,0;
%C 0,0,0,0,4,0;
%C Let M(t) = exp(t*T) = limit [1 + t*T/n]^n as n tends to infinity.
%C Then M(1) = the lower triangular padded Pascal matrix A097805, with inverse M(-1).
%C 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).
%C See A132440 for an analog and more general discussion.
%H P. Blasiak and P. Flajolet, <a href="http://arxiv.org/abs/1010.0354">Combinatorial models of creation-annihilation</a>
%H T. 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 T. Copeland, <a href="http://tcjpn.files.wordpress.com/2008/06/mathemagicalforestswp.pdf">Mathemagical Forests</a>
%H T. Copeland, <a href="http://tcjpn.files.wordpress.com/2011/06/addendum-to-mathemagical-forests-v2.pdf">Addendum to Mathemagical Forests</a>
%H G. Dattoli, B. Germano, M. Martinelli, and P. Ricci, <a href="http://arxiv.org/abs/1010.5934">Touchard like polynomials and generalized Stirling polynomials</a>
%H W. Lang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Lang/lang.html">Combinatorial interpretation of generalized Stirling numbers</a>
%F 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 1) b(0) = 0, b(1) = 0, b(n) = (n-1) * a(n-1),
%F 2) B(x) = x^2D A(x)= x (xDx)(1/x)A(x) = x^2 * Lag(1,-:xD:) A(x)/x , or
%F 3) EB(x) = D^(-1)xD EA(x),
%F 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.
%F So the exponentiated operator can be characterized as
%F 4) exp(t*T) A(x) = exp(t*x^2D) A(x) = x exp(t*xDx)(1/x)A(x)
%F = x [sum(n=0,1,...) (t*x)^n * Lag(n,-:xD:)] A(x)/x
%F = x [exp{[t*u/(1-t*u)]*:xD:} / (1-t*u) ] A(x)/x (eval. at u=x)
%F = A[x/(1-t*x)], a special Moebius or linear fractional trf.,
%F 5) exp(t*T) EA(x) = D^(-1) exp(t*x)D EA(x), a shifted Euler trf.
%F for an e.g.f., or
%F 6) [exp(t*T) * a]_n = [M(t) * a]_n
%F = [sum(k=0,...,n-1) binomial(n-1,k)* t^(n-1-k) * a(k+1)] with [M(t) * a]_0 = a_0
%F For generalizations and more on the operator x^2D, see A132440 and the references therein and above, and A094638.
%t Table[PadLeft[{n-1, 0}, n+1], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Apr 30 2014 *)
%K easy,tabl,nonn
%O 0,9
%A _Tom Copeland_, Oct 24 2012
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