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[Edit Jan 11, 2021: This is the Todd operator, whose relation to combinatorics, volumes of polytopes, summation/trace formulas, and the calculus in general is indicated with references in answers and comments to this MO-Q and this one. Also used by Hirzebruch to construct the Todd (characteristic) class.]

5. The raising op for the Swiss-knife polynomials A119879, from which the Bernoulli, Gennochi, Euler, tangent, and Springer numbers can be computed.

Note that the matrix of coefficients is a signed, masked Pascal triangle.

6. Its umbral inverse is A119467, the same masked Pascal triangle unsigned, with

[Edit Jan 11, 2021: This is the Todd operator (mod signs), whose relation to combinatorics, volumes of polytopes, summation/trace formulas, and the calculus in general is indicated with references in answers and comments to this MO-Q and this one. Also used by Hirzebruch to construct the Todd (characteristic) class/genus.]

5. The raising op for the Swiss-knife polynomials A119879, from which the Bernoulli, Gennochi, Euler, tangent, and Springer numbers can be computed

6. Its umbral inverse is A119467, a masked Pascal triangle, with

$R = x - \exp[\omega. D] $

$R = x + \exp[\omega. D].$

$R = x - D_t \; \frac{t}{e^t-1} \; |_{t=D_x} = x - \exp[\omega. D_x] $

$R = x + D_t \; \frac{t}{e^t-1} \; |_{t=D_x} = x + \exp[\omega. D_x].$

$$\log[\zeta(1+xD_x)] \delta(x-1)=\sum_p \sum_{n>0}\frac{1}{n} \delta(x-p^n),$$

1. The raising op for the reversed face polynomials of the simplices normalized by an integer, $$p_n(x)=\frac{(x+1)^{n+1}-x^{n+1}}{n+1}$$

$R = x - \exp[-\tfrac{b_{n+1}}{n+1}D] = x - \exp[\zeta(-n)D].$

$R = x + \exp[-\tfrac{b_{n+1}}{n+1}D] = x + \exp[\zeta(-n)D].$

$$\frac{J(x)}{dx} =\log[\zeta(1+xD_x)] \delta(x-1)=\sum_p \sum_{n>0}\frac{1}{n} \delta(x-p^n),$$

1. The raising op for the reversed face polynomials of the simplices normalized by an integer,

$$p_n(x)=\frac{(x+1)^{n+1}-x^{n+1}}{n+1}$$

$R = x - \exp[\omega. D] $

[1/16/21, corrected notation with $(\omega.)^n = \omega_n =-\tfrac{b_{n+1}}{n+1} = \zeta(-n) $]

$R = x + \exp[\omega. D].$