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A132681
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Infinitesimal generator matrix for a diagonally-shifted Pascal matrix, binomial(n+m,k+m), for m=1, related to Laguerre(n,x,m).
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3
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0, 2, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Analogous to the infinitesimal Pascal matrix (m=0), A132440.
In general the matrix T begins (here m=1)
0;
m+1,0;
0, m+2, 0;
0, 0, m+3, 0;
0, 0, 0, m+4, 0;
Let LM(t) = exp(t*T) = limit [1 + t*T/n]^n as n tends to infinity.
Laguerre matrix(m) = [bin(n+m,k+m)] = LM(1) = exp(T) = [ revert of A074909 for m=1 ]. Truncating the series gives the n X n submatrices. In fact, the submatrices of T are nilpotent with [Tsub_n]^(n+1) = 0 for n=0,1,2,....
Inverse Lag matrix(m) = LM(-1) = exp(-T)
Umbrally LM[b(.)] = exp(b(.)*T) = [ bin(n+m,k+m) * b(n-k) ]
A(j) = T^j / j! equals the matrix [bin(n+m,k+m) * 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 Laguerre(m) matrix. Hence the A(j)'s form a linearly independent basis for all matrices of the form [bin(n+m,k+m) d(n-k)].
For sequences with b(0) = 1, umbrally,
LM[b(.)] = exp(b(.)*T) = [ bin(n+m,k+m)] * b(n-k) ] .
[LM[b(.)]]^(-1) = exp(c(.)*T) = [ bin(n+m,k+m)] * c(n-k) ] where c = LPT(b) with LPT the list partition transform of A133314. Or,
[LM[b(.)]]^(-1) = exp[LPT(b(.))*T] = LPT[LM(b(.))] = LM[LPT(b(.))] = LM[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).
1) b(0) = 0, b(n) = (n+m) * a(n-1),
2) B(x) = x^(-m) (xDx) x^m A(x)
3) B(x) = x * Lag(1,-:xD:,m) A(x) = x * [(m+1) + xD] A(x)
4) EB(x) = D^(m) * (x) * D^(-m) EA(x) where D is the derivative w.r.t. x, (:xD:)^j = x^j*D^j, Lag(n,x,m) is the associated Laguerre polynomial and D^(-m) x^n / n! = x^(m+n) / (m+n)! are Riemann-Liouville integrals.
So the exponentiated operator can be characterized (with loose notation) as
5) exp(t*T) A(x) = x^(-m) exp(t*xDx) x^m A(x) = [sum(n=0,1,...) (t*x)^n * Lag(n,-:xD:m)] A(x) = [exp{[t*u/(1-t*u)]*:xD:} / (1-t*u)^(m+1) ] A(x) (eval. at u=x) = A[x/(1-t*x)]/(1-t*x)^(m+1), a generalized Euler transformation for an o.g.f.,
6) exp(t*T) EA(x) = D^(m) exp(t*x) D^(-m) EA(x) = [D/(D-1)]^m exp[(t+a(.))*x] = exp(t*x) [(t+D)/D]^m EA(x)
7) exp(t*T) * a = LM(t) * a = [sum(k=0,...,n) bin(n+m,k+m)* t^(n-k) * a(k)], a vector array.
With t=1 and a(k) = (-x)^k / k!, then LM(1) * a = [Laguerre(n,x,m)], a vector array with index n and the o.g.f. A(x) becomes transformed into the e.g.f. for the associated Laguerre polynomials of order m.
The exponential formulae can be umbrally extended as in A132440. And, the formulae can be extended to non-integer m.
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CROSSREFS
| Sequence in context: A169776 A091731 A127648 * A132825 A049597 A175676
Adjacent sequences: A132678 A132679 A132680 * A132682 A132683 A132684
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KEYWORD
| easy,nonn
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AUTHOR
| Tom Copeland (tcjpn(AT)msn.com), Nov 15 2007, Nov 16 2007, Nov 27 2007
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