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,).

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.

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.)

Jan. 31, 2016: With $\omega_1 = q$ and $\omega_2 = 1/q$ and no other restrictions on these variables, the expansion for the inverse function $\frac{df^{-1}(\omega)}{d\omega}$ in these notes  gives an ordinary generating function for the series $a(n) = [n+1]_q = \frac{q^{n+1}-q^{-(n+1)}}{q-q^{-1}}$, the so-called quantum numbers or integers of quantum group theory. (See OEIS-A010892 for a specific example of $q$ and the definition of quantum  numbers/integers in Knot polynomial identities and quantum group coincidences by Morison, Peters, and Snyder and in A minus sign …. (Two constructions of Jones polynomials) by Tingley.)

Google Fibonacci oscillators  (Borzov, Marinho, etc.)

See also references in the previous entry on the Kervaire-Milnor formula related to elliptic cohomology/genera.

Blog piece  on the Schwarzian derivative by Lamington

For a resume on research on the KdV equation, see “The history of q-calculus and a new method” by Ernst.

Hyperbolic expressions of polynomial sequences …” by He, Shiue, and Weng.

Continued fractions and integrable systems” by Beals, Sattinger, and Szmigielski

Elliptic cohomology and modular forms” by Landweber

Periodic cohomology theories defined by elliptic curves” by Landweber, Ravenel, and Stong

Symbolic computaton of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs” by Baldwin, Goktas, Hereman, Hong, Martino,  and Miller

What is … an elliptic genus?” by Ochanine

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

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.

Snakes in the Appell Orchard

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

Mellin Interpolation of Differential Ops and Associated Infinigens and Appell Polynomials: The Ordered, Laguerre, and Scherk-Witt-Lie Diff Ops

Interpolations of the derivative operator $D_x^n \; ,$ the fundamental ordered op $:xD_x:^n=x^nD_x^n \; ,$ the Laguerre op $:D_xx:^n = D^nx^n \; ,$ the shifted Laguerre op $(xD_xx)^n = x^nD_x^nx^n \; ,$ and the generalized Scherk-Witt Lie ops $(x^{1+y}D_x)^n$ to the fractional operators $D_x^s\; , \; :xD_x:^s = x^sD_x^s \; , \; :D_xx:^s = D_x^sx^s \; , \; (xD_xx)^s = x^sD_x^sx^s \; , \;$ and $(x^{1+y}D_x)^s$ are consistently achieved using the Mellin transform of the negated e.g.f.s of the differential ops. Associated infinitesimal generators (infinigens) are then determined for each fractional op and related to the raising ops for associated Appell sequences.

Fractional Calculus, Gamma Classes, the Riemann Zeta Function, and an Appell Pair of Sequences

The background info and comments for the MSE question Lie group heuristics for a raising operator for $\displaystyle(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0}$ and the MO question Riemann zeta function at positive integers and an Appell sequence of poylnomials introduce an Appell sequence of polynomials containing the Euler-Mascheroni constant and the Riemann zeta function evaluated at the integers greater than one. The Appell sequence can be defined by its e.g.f.

$\displaystyle \exp(\beta \; p.(z)) = \exp(\beta \; z) / \beta! \; .$

The raising op for the Appell sequence

$\displaystyle R_z=z-\frac{\mathrm{d} }{\mathrm{d} \beta}ln[\beta!]\mid _{\beta=\frac{\mathrm{d} }{\mathrm{d} z}=D_z}=z-\Psi(1+D_z) \; ,$

where $\displaystyle \Psi(x)$ is the digamma or Psi function, is associated with the infinitesimal generator

$\displaystyle R_x = \log(d/dx)= \log(D_x)$

of a class of fractional integro-derivatives through a change of variables ($\displaystyle z=\log(x)$). Exponentiation gives

$\displaystyle e^{\beta \; R_x} = D_x^{\beta} \; .$