Operator  Substitution Factor  Description

(1 + (a^{+})^{2}) a  s_{λ}(x) = \frac{xcos(λ) + sin(λ)}{cos(λ) − xsin(λ)}  One-parameter group

of homographies

\frac{\sqrt{1 + (a^{+})^{2}}}{a^{+}} a s_{λ}(x) = \sqrt{x^{2} + 2λ\sqrt{1 + x^{2}} + λ^{2}} Composition of quadratic

direct and inverse functions

s_{2}(z) = e^{(e^{z}−1)} − 1; s_{3}(z) = e^{(e^{(e^{z}−1)}−1)} − 1; s_{−1}(z) = log(1 + z)  (1)

c_{n} = (\begin{matrix}2n \\ n \end{matrix})  (2)

\frac{(y \frac{d}{dx})^{n}}{n!} \frac{x^{m}}{m!} |_{x=0} = δ_{mn} \frac{y^{n}}{n!}  (3)

∑_{n,k} I(n, k)\frac{x^{n}}{n!} y^{k} = ∑_{n≥0} ∑_{k≤n} (\begin{matrix}n \\ k \end{matrix}) k^{n−k} \frac{x^{n}}{n!} y^{k} =

∑_{k≤n} ∑_{n≥k} (\begin{matrix}n \\ k \end{matrix}) k^{n−k} \frac{x^{n−k}x^{k}}{n!} y^{k} = e^{yxe^{x}}  (4)