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A238363 Coefficients for the commutator for the logarithm of the derivative operator [log(D),x^n D^n]=d[(xD)!/(xD-n)!]/d(xD) expanded in the operators :xD:^k.

%I A238363 #177 Feb 23 2023 18:55:30
%S A238363 1,-1,2,2,-3,3,-6,8,-6,4,24,-30,20,-10,5,-120,144,-90,40,-15,6,720,
%T A238363 -840,504,-210,70,-21,7,-5040,5760,-3360,1344,-420,112,-28,8,40320,
%U A238363 -45360,25920,-10080,3024,-756,168,-36,9,-362880,403200,-226800,86400,-25200,6048,-1260,240,-45,10
%N A238363 Coefficients for the commutator for the logarithm of the derivative operator [log(D),x^n D^n]=d[(xD)!/(xD-n)!]/d(xD) expanded in the operators :xD:^k.
%C A238363 Let D=d/dx and [A,B]=A·B-B·A. Then each row corresponds to the coefficients of the operators :xD:^k = x^k D^k in the expansion of the commutator [log(D),:xD:^n]=[-log(x),:xD:^n]=sum(k=0 to n-1, a(n,k) :xD:^k). The e.g.f. is derived from [log(D), exp(t:xD:)]=[-log(x), exp(t:xD:)]= log(1+t)exp(t:xD:), using the shift property exp(t:xD:)f(x)=f((1+t)x).
%C A238363 The reversed unsigned array is A111492.
%C A238363 See the mathoverflow link and link therein to an associated mathstackexchange question for other formulas for log(D). In addition, R_x = log(D) = -log(x) + c - sum[n=1 to infnty, (-1)^n 1/n :xD:^n/n!]=
%C A238363 -log(x) + Psi(1+xD) = -log(x) + c + Ein(:xD:), where c is the Euler-Mascheroni constant, Psi(x), the digamma function, and Ein(x), a breed of the exponential integrals (cf. Wikipedia). The :xD:^k ops. commute; therefore, the commutator reduces to the -log(x) term.
%C A238363 Also the n-th row corresponds to the expansion of d[(xD)!/(xD-n)!]/d(xD) = d[:xD:^n]/d(xD) in the operators :xD:^k, or, equivalently, the coefficients of x in  d[z!/(z-n)!]/dz=d[St1(n,z)]]/dz evaluated umbrally with z=St2(.,x), i.e., z^n replaced by St2(n,x), where St1(n,x) and St2(n,x) are the signed and unsigned Stirling polynomials of the first (A008275) and second (A008277) kinds. The derivatives of the unsigned St1 are A028421. See examples. This formalism follows from the relations between the raising and lowering operators presented in the MathOverflow link and the Pincherle derivative. The results can be generalized through the operator relations in A094638, which are related to the celebrated Witt Lie algebra and pseudodifferential operators / symbols, to encompass other integral arrays.
%C A238363 A002741(n)*(-1)^(n+1) (row sums), A002104(n)*(-1)^(n+1) (alternating row sums). Column sequences: A133942(n-1), A001048(n-1), A238474, ... - _Wolfdieter Lang_, Mar 01 2014
%C A238363 Add an additional head row of zeros to the lower triangular array and denote it as T (with initial indexing in columns and rows being 0). Let dP = A132440, the infinitesimal generator for the Pascal matrix, and I, the identity matrix, then exp(T)=I+dP, i.e., T=log(I+dP). Also, (T_n)^n=0, where T_n denotes the n X n submatrix, i.e., T_n is nilpotent of order n. - _Tom Copeland_, Mar 01 2014
%C A238363 Any pair of lowering and raising ops. L p(n,x) = n·p(n-1,x) and R p(n,x) = p(n+1,x) satisfy [L,R]=1 which implies (RL)^n = St2(n,:RL:), and since (St2(·,u))!/(St2(·,u)-n)!= u^n, when evaluated umbrally, d[(RL)!/(RL-n)!]/d(RL) = d[:RL:^n]/d(RL) is well-defined and gives A238363 when the LHS is reduced to a sum of :RL:^k terms, exactly as for L=d/dx and R=x above. (Note that R_x above is a raising op. different from x, with associated L_x=-xD.) - _Tom Copeland_, Mar 02 2014
%C A238363 For relations to colored forests, disposition of flags on flagpoles, and the colorings of the vertices of the complete graphs K_n, encoded in their chromatic polynomials, see A130534. - _Tom Copeland_, Apr 05 2014
%C A238363 The unsigned triangle, omitting the main diagonal, gives A211603. See also A092271. Related to the infinitesimal generator of A008290. - _Peter Bala_, Feb 13 2017
%H A238363 Tom Copeland, <a href="http://mathoverflow.net/questions/111165/">Riemann zeta function at positive integers and an Appell sequence of polynomials</a>, 2012.
%H A238363 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>, 2014.
%H A238363 Tom Copeland, <a href="http://tcjpn.wordpress.com/2014/12/07/bernoulli-and-blissard-meet-stirling-and-pascal/">Bernoulli, Blissard, and Lie meet Stirling and the simplices: State number operators and normal ordering</a>, 2014.
%H A238363 Tom Copeland, <a href="http://tcjpn.wordpress.com/2016/11/06/compositional-inverse-operators-and-sheffer-sequences/">Compositional inverse operators and Sheffer sequences</a>, 2016.
%F A238363 a(n,k) = (-1)^(n-k-1)*n!/((n-k)*k!) for k=0 to (n-1).
%F A238363 E.g.f.: log(1+t)*exp(x*t).
%F A238363 E.g.f.for unsigned array: -log(1-t)*exp(x*t).
%F A238363 The lowering op. for the row polynomials is L=d/dx, i.e., L p(n,x) = n*p(n-1,x).
%F A238363 An e.g.f. for an unsigned related version is -log(1+t)*exp(x*t)/t= exp(t*s(·,x)) with s(n,x)=(-1)^n * p(n+1,-x)/(n+1). Let L=d/dx and R= x-(1/((1-D)log(1-D))+1/D),then R s(n,x)= s(n+1,x) and L s(n,x)= n*s(n-1,x), defining a special Sheffer sequence of polynomials, an Appell sequence. So, R (-1)^(n-1) p(n,-x)/n = (-1)^n p(n+1,-x)/(n+1).
%F A238363 From _Tom Copeland_, Apr 17 2014: (Start)
%F A238363 Dividing each diagonal by its first element (-1)^(n-1)*(n-1)! yields the reverse of A104712.
%F A238363 Multiply each n-th diagonal of the Pascal lower triangular matrix by x^n and designate the result as A007318(x) = P(x). Then with dP = A132440, M = padded A238363 = A238385-I, I = identity matrix, and (B(.,x))^n = B(n,x) = the n-th Bell polynomial Bell(n,x) of A008277,
%F A238363 A) P(x)= exp(x*dP) = exp[x*(e^M-I)] = exp[M*B(.,x)] = (I+dP)^B(.,x), and
%F A238363 B) P(:xD:)=exp(dP:xD:)=exp[(e^M-I):xD:]=exp[M*B(.,:xD:)]=exp[M*xD]=
%F A238363   (1+dP)^(xD) with action P(:xD:)g(x) = exp(dP:xD:)g(x) = g[(I+dP)*x].
%F A238363 C) P(x)^m = P(m*x). P(2x) = A038207(x) = exp[M*B(.,2x)], face vectors of n-D hypercubes. (End)
%F A238363 From _Tom Copeland_, Apr 26 2014: (Start)
%F A238363 M = padded A238363 = A238385-I
%F A238363 A)  = [St1]*[dP]*[St2] = [padded A008275]*A132440*A048993
%F A238363 B)  = [St1]*[dP]*[St1]^(-1)
%F A238363 C)  = [St2]^(-1)*[dP]*[St2]
%F A238363 D)  = [St2]^(-1)*[dP]*[St1]^(-1),
%F A238363   where [St1]=padded A008275 just as [St2]=A048993=padded A008277.
%F A238363 E) P(x) = [St2]*exp(x*M)*[St1] = [St2]*(I + dP)^x*[St1].
%F A238363 F) exp(x*M) = [St1]*P(x)*[St2] = (I + dP)^x,
%F A238363   where (I + dP)^x = sum(k>=0, C(x,k)*dP^k).
%F A238363 Let 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) and W(y) := V(x.) evaluated umbrally with (x.)^n = x_n = (y)_n = y!/(y-n)!. Then
%F A238363 G) U(:xD:) = dV(:xD:)/d(xD) = dW(xD)/d(xD) evaluated with (xD)^n = Bell(n,:xD:),
%F A238363 H) U(x) = dV(x.)/dy := dW(y)/dy evaluated with y^n=y_n=Bell(n,x), and
%F A238363 I) U(x) = Rv * M * Cv(x).   (Cf. A132440, A074909.) (End)
%F A238363 The Bernoulli polynomials Ber_n(x) are related to the polynomials q_n(x) = p(n+1,x) / (n+1) with the e.g.f. [log(1+t)/t] e^(xt) (cf. s_n (x) above) as Ber_n(x) = St2_n[q.(St1.(x))], umbrally, or [St2]*[q]*[St1], in matrix form. Since q_n(x) is an Appell sequence of polynomials, q_n(x) = [log(1+D_x)/D_x]x^n. - _Tom Copeland_, Nov 06 2016
%e A238363 The first few row polynomials are
%e A238363 p(1,x)=  1
%e A238363 p(2,x)= -1 + 2x
%e A238363 p(3,x)=  2 - 3x + 3x^2
%e A238363 p(4,x)= -6 + 8x - 6x^2 + 4x^3
%e A238363 p(5,x)= 24 -30x +20x^2 -10x^3 + 5x^4
%e A238363 ...........
%e A238363 For n=3: z!/(z-3)!=z^3-3z^2+2z=St1(3,z) with derivative 3z^2-6z+2, and
%e A238363 3·St2(2,x)-6·St2(1,x)+2=3(x^2+x)-6x+2=3x^2-3x+2=p(3,x). To see the relation to the operator formalism, note that (xD)^k=St2(k,:xD:) and (xD)!/(xD-k)!=[St2(·,:xD:)]!/[St2(·,:xD:)-k]!= :xD:^k.
%e A238363 The triangle a(n,k) begins:
%e A238363 n\k       0       1       2      3      4     5      6    7   8   9 ...
%e A238363 1:        1
%e A238363 2:       -1       2
%e A238363 3:        2      -3       3
%e A238363 4:       -6       8      -6      4
%e A238363 5:       24     -30      20    -10      5
%e A238363 6:     -120     144     -90     40    -15     6
%e A238363 7:      720    -840     504   -210     70   -21      7
%e A238363 8:    -5040    5760   -3360   1344   -420   112    -28    8
%e A238363 9:    40320  -45360   25920 -10080   3024  -756    168  -36   9
%e A238363 10: -362880  403200 -226800  86400 -25200  6048  -1260  240 -45  10
%e A238363 ... formatted by _Wolfdieter Lang_, Mar 01 2014
%e A238363 -----------------------------------------------------------------------
%t A238363 a[n_, k_] := (-1)^(n-k-1)*n!/((n-k)*k!); Table[a[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Jul 09 2015 *)
%Y A238363 Cf. A094587, A132013, A132014, A132159, A235706, A238385.
%Y A238363 Cf. A008290, A092271, A211603.
%K A238363 sign,tabl
%O A238363 1,3
%A A238363 _Tom Copeland_, Feb 25 2014
%E A238363 Pincherle formalism added by _Tom Copeland_, Feb 27 2014