OFFSET
0,5
COMMENTS
Let M(t) = exp(t*T) = lim_{n->oo} (1 + t*T/n)^n.
Pascal matrix = [ binomial(n,k) ] = M(1) = exp(T), truncating the series gives the n X n submatrices.
Inverse Pascal matrix = M(-1) = exp(-T) = matrix for inverse binomial transform.
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.
For sequences with b(0) = 1, umbrally,
M[b(.)] = exp(b(.)*T) = [ binomial(n,k) * b(n-k) ] = matrices associated to b by LPT.
[M[b(.)]]^(-1) = exp(c(.)*T) = [ binomial(n,k) * c(n-k) ] = matrices associated to c, where c = LPT(b) . Or,
[M[b(.)]]^(-1) = exp[LPT(b(.))*T] = LPT[M(b(.))] = M[LPT(b(.))]= M[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.
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.
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).
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.
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.
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(0) = 0, b(n) = n * a(n-1),
2) B(x) = xDx A(x)
3) B(x) = x * Lag(1,-:xD:) A(x)
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.
So the exponentiated operator can be characterized as
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.,
6) exp(t*T) EA(x) = exp(t*x)*EA(x) = exp[(t+a(.))*x], gen. Euler trf. for an e.g.f.
7) exp(t*T) * a = M(t) * a = [Sum_{k=0..n} binomial(n,k) * t^(n-k) * a(k)].
The umbral extension of formulas 5, 6 and 7 gives formally
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,
9) exp[c(.)*T] EA(x) = exp(c(.)*x)*EA(x) = exp[(c(.)+a(.))*x]
10) exp[c(.)*T] * a = M[c(.)] * a = [Sum_{k=0..n} binomial(n,k) * c(n-k) * a(k)] .
The n X n principal submatrix of T is nilpotent, in particular, [Tsub_n]^(n+1) = 0, n=0,1,2,3,....
Note (xDx)^n = x^n D^n x^n = x^n n! (:Dx:)^n/n! = x^n n! Lag(n,-:xD:).
The operator xDx is an important, classical operator explored by among others Dattoli, Al-Salam, Carlitz and Stokes and even earlier investigators.
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
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
From Tom Copeland, Apr 25 2014: (Start)
Conjugation or "similarity" transformations of [dP]=A132440 have an operator interpretation (cf. A074909 and A238363):
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.
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.
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
A) [F2]*[dP]*[F1]
B) = [F2]*[dP]*[F2]^(-1)
C) = [F1]^(-1)*[dP]*[F1],
where [F1] is the lower triangular matrix with the n-th row the coefficients of F1(n,x) and analogously for [F2].
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).
D) dV(B)/dA = Rv * [F2]*[dP]*[F1] * Cv(B).
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)
REFERENCES
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).
LINKS
Robert Israel, Table of n, a(n) for n = 0..10000
W. A. Al-Salam, Operational representations for the Laguerre and other polynomials, Duke Math. Jour., vol. 31 (1964), pp. 127-142.
Tom Copeland, Fractional Calculus, Gamma Classes, the Riemann Zeta Function, and an Appell Pair of Sequences.
Tom Copeland, Goin' with the Flow: Logarithm of the Derivative Operator.
Tom Copeland, The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions (pdf).
Tom Copeland, Mathemagical Forests.
G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009. (Cf. Viennot's Laguerre histoires.)
K. A. Penson, P. Blasiak, A. Horzela, G.H.E. Duchamp and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, arXiv:0904.0369 [math-ph], 2009.
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.
G. Stokes, Note on certain formulae in the calculus of operations, Proceedings of the Royal Society of Edinburgh, IX, pp. 101-102, 1876.
FORMULA
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
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
From Tom Copeland, Apr 26 2014: (Start)
A) T = exp(A238385-I) - I
B) = [St1]*P*[St2] - I
C) = [St1]*P*[St1]^(-1) - I
D) = [St2]^(-1)*P*[St2] - I
E) = [St2]^(-1)*P*[St1]^(-1) - I
where P=A007318, [St1]=padded A008275 just as [St2]=A048993=padded A008277, and I=identity matrix. (End)
From Robert Israel, Oct 02 2015: (Start)
G.f. Sum_{k >= 1} k x^((k+3/2)^2/2 - 17/8) is related to Jacobi theta functions.
If 8*n+17 = y^2 is a square, then a(n) = (y-3)/2, otherwise a(n) = 0. (End)
EXAMPLE
Matrix T begins
0;
1,0;
0,2,0;
0,0,3,0;
0,0,0,4,0;
...
MAPLE
seq(op([0$i, i]), i=1..20); # Robert Israel, Oct 02 2015
MATHEMATICA
Table[PadLeft[{n, 0}, n+1], {n, 0, 11}] // Flatten (* Jean-François Alcover, Apr 30 2014 *)
CROSSREFS
KEYWORD
AUTHOR
Tom Copeland, Nov 13 2007, Nov 15 2007, Nov 22 2007, Dec 02 2007
EXTENSIONS
Missing zero added in table by Tom Copeland, Feb 25 2014
STATUS
approved