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}.

Errata:

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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One Response to The Creation / Raising Operators for Appell Sequences

  1. Pingback: The Pincherle Derivative and the Appell Raising Operator | Shadows of Simplicity

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