%I A132440
%S A132440 0,1,0,0,2,0,0,0,3,0,0,0,0,4,0,0,0,0,0,5,0,0,0,0,0,6,0,0,0,0,0,0,0,7,0,
%T A132440 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,
%U A132440 0,0,0,0,0,11,0
%N A132440 Infinitesimal Pascal matrix: generator (lower triangular matrix representation) 
               of the Pascal matrix, the classical operator xDx, iterated Laguerre 
               transforms, associated matrices of the list partition transform and 
               general Euler transformation for sequences.
%C A132440 Matrix T begins
%C A132440 0;
%C A132440 1,0;
%C A132440 0,2,0;
%C A132440 0,0,3,0;
%C A132440 0,0,0,4,0;
%C A132440 Let M(t) = exp(t*T) = limit [1 + t*T/n]^n as n tends to infinity.
%C A132440 Pascal matrix = [ binomial(n,k) ] = M(1) = exp(T), truncating the series 
               gives the n X n submatrices.
%C A132440 Inverse Pascal matrix = M(-1) = exp(-T) = matrix for inverse binomial 
               transform.
%C A132440 A(j) = T^j / j! equals the matrix [bin(n,k) * 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 Pascal triangle. Hence 
               the A(j)'s form a linearly independent basis for all matrices of 
               the form [binomial(n,k) d(n-k)] which include as a subset the invertible 
               associated matrices of the list partition transform (LPT) of A133314.
%C A132440 For sequences with b(0) = 1, umbrally,
%C A132440 M[b(.)] = exp(b(.)*T) = [ binomial(n,k) * b(n-k) ] = matrices associated 
               to b by LPT.
%C A132440 [M[b(.)]]^(-1) = exp(c(.)*T) = [ binomial(n,k) * c(n-k) ] = matrices 
               associated to c, where c = LPT(b) . Or,
%C A132440 [M[b(.)]]^(-1) = exp[LPT(b(.))*T] = LPT[M(b(.))] = M[LPT(b(.))] = M[c(.)] 
               .
%C A132440 This is related to xDx, the iterated Laguerre transform and the general 
               Euler transformation of a sequence through the comments in A132013 
               and A132014 and the relation [sum(k=0,...,n) binomial(n,k) * b(n-k) 
               * d(k)] = M(b)*d, (n-th term). See also A132382.
%C A132440 If b(n,x) is a binomial type Sheffer sequence, then M[b(.,x)]*s(y) = 
               s(x+y) when s(y) = (s(0,y),s(1,y),s(2,y),...) is an array for a Sheffer 
               sequence with the same delta operator as b(n,x) and [M[b(.,x)]]^(-1) 
               is given by the formulae above with b(n) replaced by b(n,x) as b(0,
               x)=1 for a binomial type Sheffer sequence.
%C A132440 T = I - A132013 and conversely A132013 = I - T, which is the matrix representation 
               for the iterated mixed order Laguerre transform characterized in 
               A132013 (and A132014).
%C A132440 (I-T)^m generates the group [A132013]^m for m= 0,1,2,.. discussed in 
               A132014.
%C A132440 The inverse is 1/(I-T) = I+T+T^2+T^3+... = [A132013]^(-1) = A094587 with 
               the associated sequence (0!,1!,2!,3!,...) under the LPT.
%C A132440 And 1/(I-T)^2 = I+2*T+3*T^2+4*T^3+... = [A132013]^(-2) = A132159 with 
               the associated sequence (1!,2!,3!,4!,...) under the LPT.
%C A132440 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).
%C A132440 1) b(0) = 0, b(n) = n * a(n-1),
%C A132440 2) B(x) = xDx A(x)
%C A132440 3) B(x) = x * Lag(1,-:xD:) A(x)
%C A132440 4) EB(x) = x * EA(x) where D is the derivative w.r.t. x, (:xD:)^j = x^j*D^j 
               and Lag(n,x) is the Laguerre polynomial.
%C A132440 So the exponentiated operator can be characterized as
%C A132440 5) exp(t*T) A(x) = exp(t*xDx) A(x) = [sum(n=0,1,...) (t*x)^n * Lag(n,
               -:xD:)] A(x) = [exp{[t*u/(1-t*u)]*:xD:} / (1-t*u) ] A(x) (eval. at 
               u=x) = A[x/(1-t*x)]/(1-t*x), a generalized Euler transformation for 
               an o.g.f.,
%C A132440 6) exp(t*T) EA(x) = exp(t*x)*EA(x) = exp[(t+a(.))*x], gen. Euler trf. 
               for an e.g.f.
%C A132440 7) exp(t*T) * a = M(t) * a = [sum(k=0,...,n) binomial(n,k) * t^(n-k) 
               * a(k)] .
%C A132440 The umbral extension of formulae 5, 6 and 7 gives formally
%C A132440 8) exp[c(.)*T] A(x) = exp(c(.)*xDx) A(x) = [sum(n=0,1,...) (c(.)*x)^n 
               * Lag(n,-:xD:)] A(x) = [exp{[c(.)*u/(1-c(.)*u)]*:xD:} / (1-c(.)*u) 
               ] A(x) (eval. at u=x) = A[x/(1-c(.)*x)]/(1-c(.)*x), where the umbral 
               evaluation should be applied only after a power series in c is obtained,
%C A132440 9) exp[c(.)*T] EA(x) = exp(c(.)*x)*EA(x) = exp[(c(.)+a(.))*x]
%C A132440 10) exp[c(.)*T] * a = M[c(.)] * a = [sum(k=0,...,n) binomial(n,k) * c(n-k) 
               * a(k)] .
%C A132440 The n X n principal submatrix of T is nilpotent, in particular, [Tsub_n]^(n+1) 
               = 0, n=0,1,2,3,....
%C A132440 Note (xDx)^n = x^n D^n x^n = x^n n! (:Dx:)^n/n! = x^n n! Lag(n,-:xD:) 
               .
%C A132440 The operator xDx is an important, classical operator explored by among 
               others Dattoli, Al-Salam, Carlitz and Stokes and even earlier investigators.
%C A132440 Contribution from Karol A. Penson (penson(AT)lptl.jussieu.fr), Sep 15 
               2009: For a recent treatment of xDx, DxD and more general operators 
               see the paper "Laguerre-type derivatives: Dobinski relations and 
               combinatorial identities"
%D A132440 K. A. Penson, P. Blasiak, A. Horzela, G.H.E. Duchamp and A. I. Solomon, 
               "Laguerre-type derivatives: Dobinski relations and combinatorial 
               identities", Journal of Mathematical Physics vol.50, (2009) 083512 
               and arXiv:0904.0369.
%Y A132440 Sequence in context: A083927 A154724 A134402 this_sequence A127647 A140579 
               A091227
%Y A132440 Adjacent sequences: A132437 A132438 A132439 this_sequence A132441 A132442 
               A132443
%K A132440 easy,nonn
%O A132440 0,5
%A A132440 Tom Copeland (tcjpn(AT)msn.com), Nov 13 2007, Nov 15 2007, Nov 22 2007, 
               Dec 02 2007