## The Creation / Raising Operators for Appell Sequences

The Creation / Raising Operators for Appell Sequences is a pdf presenting representations 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 class discussed by Libgober and Lu.

## The Elliptic Lie Triad: KdV and Ricatti Equations, Infinigens, and Elliptic Genera

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

This site is not correctly updating, so the notes have been transcribed to this pdf.

## 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

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

## A Class of Differential Operators and the Stirling Numbers

The differential operator $\displaystyle (x^{1+y} \; D)^n$ with $\displaystyle D=d/dx$ can easily be expanded in terms of the operators $\displaystyle (:xD:)^n = x^n \; D^n$ by considering its action on $\displaystyle x^s \; .$