The Creation / Raising Operators for Appell Sequences is a pdf presenting representations of the raising operator and its exponentiation 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.
This site is not correctly updating, so the notes have been transcribed to this pdf.
The Euler-Bernoulli numbers: what they count and associations to algebraic geometry, elliptic curves, and differential ops. Coming soon.
The background info and comments for the MSE question Lie group heuristics for a raising operator for 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.
The raising op for the Appell sequence
where is the digamma or Psi function, is associated with the infinitesimal generator
of a class of fractional integro-derivatives through a change of variables (). Exponentiation gives
The differential operator with can easily be expanded in terms of the operators by considering its action on