A132440 - OEIS

%I #107 Sep 29 2023 19:53:59

%S 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 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 0,0,0,0,0,0,0,0,11,0

%N 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 Let M(t) = exp(t*T) = lim_{n->oo} (1 + t*T/n)^n.

%C Pascal matrix = [ binomial(n,k) ] = M(1) = exp(T), truncating the series gives the n X n submatrices.

%C Inverse Pascal matrix = M(-1) = exp(-T) = matrix for inverse binomial transform.

%C A(j) = T^j / j! equals the matrix [binomial(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 For sequences with b(0) = 1, umbrally,

%C M[b(.)] = exp(b(.)*T) = [ binomial(n,k) * b(n-k) ] = matrices associated to b by LPT.

%C [M[b(.)]]^(-1) = exp(c(.)*T) = [ binomial(n,k) * c(n-k) ] = matrices associated to c, where c = LPT(b) . Or,

%C [M[b(.)]]^(-1) = exp[LPT(b(.))*T] = LPT[M(b(.))] = M[LPT(b(.))]= M[c(.)].

%C 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 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 formulas above with b(n) replaced by b(n,x) as b(0,x)=1 for a binomial-type Sheffer sequence.

%C 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 (I-T)^m generates the group [A132013]^m for m = 0,1,2,... discussed in A132014.

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

%C 2) B(x) = xDx A(x)

%C 3) B(x) = x * Lag(1,-:xD:) A(x)

%C 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 So the exponentiated operator can be characterized as

%C 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 6) exp(t*T) EA(x) = exp(t*x)*EA(x) = exp[(t+a(.))*x], gen. Euler trf. for an e.g.f.

%C 7) exp(t*T) * a = M(t) * a = [Sum_{k=0..n} binomial(n,k) * t^(n-k) * a(k)].

%C The umbral extension of formulas 5, 6 and 7 gives formally

%C 8) exp[c(.)*T] A(x) = exp(c(.)*xDx) A(x) = [Sum_{n>=0} (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 9) exp[c(.)*T] EA(x) = exp(c(.)*x)*EA(x) = exp[(c(.)+a(.))*x]

%C 10) exp[c(.)*T] * a = M[c(.)] * a = [Sum_{k=0..n} binomial(n,k) * c(n-k) * a(k)] .

%C The n X n principal submatrix of T is nilpotent, in particular, [Tsub_n]^(n+1) = 0, n=0,1,2,3,....

%C Note (xDx)^n = x^n D^n x^n = x^n n! (:Dx:)^n/n! = x^n n! Lag(n,-:xD:).

%C The operator xDx is an important, classical operator explored by among others Dattoli, Al-Salam, Carlitz and Stokes and even earlier investigators.

%C For a recent treatment of xDx, DxD and more general operators see the paper "Laguerre-type derivatives: Dobinski relations and combinatorial identities". - _Karol A. Penson_, Sep 15 2009

%C See Copeland's link for generalized Laguerre functions and connection to fractional differ-integrals in exercises through (:Dx:)^a/a!=(D^a x^a)/a!. - _Tom Copeland_, Nov 17 2011

%C From _Tom Copeland_, Apr 25 2014: (Start)

%C Conjugation or "similarity" transformations of [dP]=A132440 have an operator interpretation (cf. A074909 and A238363):

%C In general, select two operators A and B such that A^n = F1(n,B) and B^n = F2(n,A); then A^n = F1(n,F2(.,A)) and B^n = F2(n,F1(.,B)), evaluated umbrally, i.e., F1(n,F2(.,x))=F2(n,F1(.,x))=x^n, implying the polynomials F1 and F2 are an umbral compositional inverse pair.

%C One such pair are the Bell polynomials Bell(n,x) and falling factorials (x)_n with Bell(n,:xD:)=(xD)^n and (xD)_n=:xD:^n (cf. A074909). Another are the Laguerre polynomials LN(n,x)= n!*Lag(n,x) (A021009), which are umbrally self-inverse, with LN(n,-:xD:)=:Dx:^n and LN(n,:Dx:)= (-:xD:)^n with :Dx:^n=D^n*x^n.

%C Evaluating, for n>=0, the operator derivative d(B^n)/dA = d(F2(n,A))/dA in the basis B^n, i.e., with A^n finally replaced by F1(n,B), or A^n=F1(.,B)^n=F1(n,B), is equivalent to the matrix conjugation

%C A)   [F2]*[dP]*[F1]

%C B) = [F2]*[dP]*[F2]^(-1)

%C C) = [F1]^(-1)*[dP]*[F1],

%C where [F1] is the lower triangular matrix with the n-th row the coefficients of F1(n,x) and analogously for [F2].

%C So, given the row vector Rv=(c0 c1 c2 c3 ...) and the column vector Cv(x)=(1 x x^2 x^3 ...)^Transpose, form the power series V(x)=Rv*Cv(x).

%C D) dV(B)/dA = Rv * [F2]*[dP]*[F1] * Cv(B).

%C E) With A=D and B=D, F1(n,x)=F2(n,x)=x^n and [F1]=[F2]=I. Then d(B^n)/dA = d(D^n)/dD = n * D^(n-1); therefore, consistently [F2]*[dP]*[F1] = [dP] and dV(D)/dD = Rv * [dP] * Cv(D). (End)

%D T. Mansour and M. Schork, Commutation Relations, Normal Ordering, and Stirling Numbers, Chapman and Hall/CRC, 2015, (x^n D^n x^n on p. 187).

%H Robert Israel, <a href="/A132440/b132440.txt">Table of n, a(n) for n = 0..10000</a>

%H W. A. Al-Salam, <a href="http://projecteuclid.org/euclid.dmj/1077375084">Operational representations for the Laguerre and other polynomials</a>, Duke Math. Jour., vol. 31 (1964), pp. 127-142.

%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/09/09/fractional-calculus-gamma-classes-the-riemann-zeta-function-and-an-appell-pair-of-sequences/">Fractional Calculus, Gamma Classes, the Riemann Zeta Function, and an Appell Pair of Sequences</a>.

%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2014/08/03/goin-with-the-flow-logarithm-of-the-derivative/">Goin' with the Flow: Logarithm of the Derivative Operator</a>.

%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2012/11/29/infinigens-the-pascal-pyramid-and-the-witt-and-virasoro-algebras/">Infinigens, the Pascal Pyramid, and the Witt and Virasoro Algebras</a>.

%H Tom Copeland, <a href="http://tcjpn.files.wordpress.com/2011/11/the-mt-bell-dobinski-chgf.pdf">The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions (pdf)</a>.

%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2008/06/12/mathemagical-forests/">Mathemagical Forests</a>.

%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/09/16/mellin-transform-interpolation-of-differential-operators/">Mellin Interpolation of Differential Ops and Associated Infinigens and Appell Polynomials: The Ordered, Laguerre, and Scherk-Witt-Lie Diff Ops</a>.

%H G. Hetyei, <a href="http://arxiv.org/abs/0909.4352">Meixner polynomials of the second kind and quantum algebras representing su(1,1)</a>, arXiv preprint arXiv:0909.4352 [math.QA], 2009. (Cf. Viennot's Laguerre histoires.)

%H K. A. Penson, P. Blasiak, A. Horzela, G.H.E. Duchamp and A. I. Solomon, <a href="http://arxiv.org/abs/0904.0369">Laguerre-type derivatives: Dobinski relations and combinatorial identities</a>, arXiv:0904.0369 [math-ph], 2009.

%H K. A. Penson, P. Blasiak, A. Horzela, G.H.E. Duchamp and A. I. Solomon, <a href="http://dx.doi.org/10.1063/1.3155380">Laguerre-type derivatives: Dobinski relations and combinatorial identities</a>, Journal of Mathematical Physics vol. 50, (2009) 083512.

%H G. Stokes, <a href="https://doi.org/10.1017/S0370164600031758">Note on certain formulae in the calculus of operations</a>, Proceedings of the Royal Society of Edinburgh, IX, pp. 101-102, 1876.

%F T = log(P) with the Pascal matrix P:=A007318. This should be read as T_N = log(P_N) with P_N the N X N matrix P, N>=2. Because P_N is lower triangular with all diagonal elements 1, the series log(1_N-(1_N-P_N)) stops after N-1 terms because (1_N-P_N)^N is the 0_N-matrix. - _Wolfdieter Lang_, Oct 14 2010

%F 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*L*R 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). - _Tom Copeland_, Oct 25 2012

%F From _Tom Copeland_, Apr 26 2014: (Start)

%F A) T = exp(A238385-I) - I

%F B)   = [St1]*P*[St2] - I

%F C)   = [St1]*P*[St1]^(-1) - I

%F D)   = [St2]^(-1)*P*[St2] - I

%F E)   = [St2]^(-1)*P*[St1]^(-1) - I

%F where P=A007318, [St1]=padded A008275 just as [St2]=A048993=padded A008277, and I=identity matrix. (End)

%F From _Robert Israel_, Oct 02 2015: (Start)

%F G.f. Sum_{k >= 1} k x^((k+3/2)^2/2 - 17/8) is related to Jacobi theta functions.

%F If 8*n+17 = y^2 is a square, then a(n) = (y-3)/2, otherwise a(n) = 0. (End)

%e Matrix T begins

%e   0;

%e   1,0;

%e   0,2,0;

%e   0,0,3,0;

%e   0,0,0,4,0;

%e   ...

%p seq(op([0$i,i]),i=1..20); # _Robert Israel_, Oct 02 2015

%t Table[PadLeft[{n, 0}, n+1], {n, 0, 11}] // Flatten (* _Jean-François Alcover_, Apr 30 2014 *)

%K easy,nonn,tabl

%O 0,5

%A _Tom Copeland_, Nov 13 2007, Nov 15 2007, Nov 22 2007, Dec 02 2007

%E Missing zero added in table by _Tom Copeland_, Feb 25 2014