Dirac-Appell Sequences

The Pincherle derivative [T^n(L,R),R] = \frac {d}{dL}T^n(L,R)= n \cdot T^{n-1}(L,R)  is implicitly used in Eqn. 2.19 page 13 of “Mastering the master field” by Gopakumar and Gross. The raising and creation operators in the paper are analogous to those for a Laplace-dual Appell sequence, or Dirac-Appell sequence, comprised of the Dirac delta function and its derivatives, formed by taking the inverse Laplace transform of the polynomials of an Appell polynomial sequence.

The fundamental D-A sequence can be defined as the sequence \delta^{(n)}(x) with L = -x and R = D = d/dx and e.g.f. e^{tD} \delta(x) = \delta(x+t). Another example is provided by OEIS A099174 with the D-A sequence H_n(x) = h_n(D) \delta(x) where h_n(x) are the modified Hermite polynomials listed in the Example section of the entry. The modified Hermite polynomials can be characterized several ways:

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The Pincherle Derivative and the Appell Raising Operator

The raising and lowering operators R and L for a sequence of functions \psi_n(x), with n= 0,1, 2, ... and \psi_0(x)=1, defined by

R \; \psi_n(x) = \psi_{n+1}(x) and L \; \psi_n(x) = n \; \psi_{n-1}(x)

have the commutator relation

[L,R] = LR-RL = 1

with respect to action on the space spanned by this sequence of functions.

If for m any particular natural number

[L^m,R] = m \; L^{m-1} = \frac{d}{dL}L^m ,


mL^m =L \; [L^m,R] = L^{m+1}R - LRL^{m}

= L^{m+1}R - (1+RL)L^{m} = L^{m+1}R - RL^{m+1} - L^m ,


[L^{m+1},R] = (m+1) L^{m} = \frac{d}{dL}L^{m+1}.

Since this holds for m=1, the relation holds for all natural numbers, and formally for a function f(x)=e^{a.x} analytic about the origin (or a formal power series or exponential  generating function)

[f(L),R] = [e^{a.L},R]= \frac{d}{dL}e^{a.L} = a. \; e^{a.L}=\frac{d}{dL}f(L).

The reader should be able to modify the argument to show the dual relation

[L,f (R)] = \frac {d}{dR}f(R) .

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Generators, Inversion, and Matrix, Binomial, and Integral Transforms

Generators, Inversion, and Matrix, Binomial, and Integral Transforms is a belated set of notes (pdf) on a derivation of a generating function for the row polynomials of  OEIS-A111999 from its relation to the compositional inversion (a Lagrange inversion formula, LIF) presented in A133932 of invertible functions represented umbrally as logarithmic series -ln(1-b.x). The results show that A111999 is a natural reduction of A133932.

Along the way, more general results are given involving the relations among Borel-Laplace transforms; compositional inversion in general; binomial transforms of rows, columns, and diagonals of matrices; infinitesimal generators; and the generating functions and reversals of binomial and Appell Sheffer polynomials, in particular the cycle index polynomials of the symmetric groups, or partition polynomials of the refined Stirling numbers of the first kind A036039.

A table has been added in Appendix III to illustrate how the analysis applies to the two other complementary LIFs A134685 , based on the refined Stirling numbers of the second kind A036040 (a refinement of the Bell / Touchard / exponential polynomials A008277), and A133437, based on the refined Lah numbers A130561 (a refinement of the Lah polynomials  A008297, A105278, normalized Laguerre polynomials of order -1).

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The Creation / Raising Operators for Appell Sequences

The Creation / Raising Operators for Appell Sequences is a pdf presenting reps of the raising operator R and its exponentiation \exp(tR) for normal and logarithmic Appell sequences of polynomials as differential and integral operators. The Riemann zeta and digamma, or Psi, function are connected to fractional calculus and associated Appell sequences for a characteristic genus discussed by Libgober and Lu.

Additional notes:

Using the inverse Mellin transform rep of the Dirac delta function given in an earlier entry leads to  the integral kernel K(x,-m) = H(1-x) \frac{(1-x)^{-m-1}}{(-m-1)!}=(-1)^m \frac{d}{dx}^m \delta(1-x)=\delta^{(m)}(1-x) for K(x,t) on page 10.

(Added 9/8/2016)  The post Bernoulli Appells contains yet another rep for an Appell polynomial raising operator:

R = e^{B.(0)D_x} \; x \; e^{\hat{B}(0)D_x} = x - x + e^{B.(0)D_x} \; x \; e^{\hat{B}(0)D_x} = x - e^{B.(0)D_x}[e^{\hat{B}(0)D_x},x] ,

which holds for any sequence of Appell polynomials B_n(x) and its umbral inverse Appell sequence \hat{B}(x). See OEIS A263634 for matrix reps of the raising op.

(Added 9/15/2016) For the convolution rep of the derivative op and its relation to the Mellin transform in Part III of the pdf, see the post Note on the Inverse Mellin Transform and the Dirac Delta Function on the inverse Mellin transform rep for the derivative of the Dirac delta :

\displaystyle \delta^{'}(y-x) = \frac{d}{dy} \delta(y-x) = \frac{d}{dy} \int_{\sigma-i \infty  }^{\sigma + \infty} y^{s-1}x^{-s}ds =\int_{\sigma-i \infty  }^{\sigma + \infty} (s-1) y^{s-2}x^{-s}ds,

so in this sense

\displaystyle \delta^{'}(1-x) = \int_{\sigma-i \infty  }^{\sigma + \infty} (s-1) x^{-s}ds = H(1-x) \frac{(1-x)^{t-1}}{(t-1)!}|_{t=-1}.


Pg. 2: A(\phi.(:xD_x:+t) should be A(\phi.(:xD_x:)+t).

Pg. 9: e^{tR_x}x^s should be e^{tR_x}.

Related Stuff:

Two applications of elementary knot theory to Lie algebras and Vassiliev invariants” by Bar-Natan, Le, Thurston (note formulas containing sinh). See also the Thurston paper referenced in the post Bernoulli Appells.

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The Elliptic Lie Triad: KdV and Riccati Equations, Infinigens, and Elliptic Genera

The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera

(This site was not correctly updating, so the notes were transcribed to this pdf.)

Addendum to The Elliptic Lie Triad

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The Kervaire-Milnor Formula

The K-M formula and its ingredients are presented in

1) Bernoulli numbers and the unity of mathematics by Barry Mazur, p.14, Secs. 4, 5, and 6

2) Differential topology forty-six years later by Milnor

3) Homotopy group of spheres Wikipedia

4) Exotic sphere Wikipedia

5) J-homomorphism Wikipedia

6) A history of the Arf-Kervaire invariant problem by Snaith

7) Bernoulli numbers, homotopy groups, and a theorem of Rohlin by Milnor and Kervaire

The K-M formula, as presented by Mazur, is

card [\Theta_{4k-1}] = R(k) \; card[ H_{4k-1}] \; B_{2k}/2k

where (if I interpret Mazur, and Milnor, correctly) \Theta_{j} is the group of homotopy spheres up to h-cobordism, or essentially the set of all oriented diffeomorphism classes of closed smooth homotopy n-spheres; R(k)=2^{2k-2}(2^{2k-1}-1) for odd k and twice that for even k; H_{j} is the group of stable homotopy classes of continuous maps from the (m+j)-sphere to the j-sphere, and B_n are the Bernoulli numbers.

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Snakes in the Appell Orchard

The Euler-Bernoulli numbers: what they count and associations to algebraic geometry, elliptic curves, and differential ops. Coming soon.

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