Name of "divided difference" transform $\frac{f(x)-f(x_0)}{x-x_0}$ and special case $\frac{e^x - 1}{x}$?

Question

Given an analytic function / formal power series

$$\displaystyle f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(x_0)}{n!}\left(x-x_{0}\right)^{n}=f(x_0)+f'(x_0)(x-x_{0})+ \tfrac{1}{2}f''(x_0)(x-x_{0})^{2}+\ldots$$

we can construct another analytical function via the "divided difference" transformation

$$ R(f)(x) := \frac{f(x)-f(x_0)}{x-x_0} := f'(x_0) + \tfrac{1}{2}f''(x_0)(x-x_0) + \tfrac{1}{6}f'''(x_0)(x-x_0)^2 + \ldots $$

Note that this is similar but not equal to the derivative operator

$$ D(f)(x) = f'(x_0) + f''(x_0)(x-x_0) + \tfrac{1}{2}f'''(x_0)(x-x_0)^2 + \ldots $$

1. Does this transformation have a name in the literature? What are its properties? In the discrete case it is known as divided differences, however here I am explicitly interested in this transform as a map $$T\colon A(D)\to A(D),\, f\mapsto R(f)$$ between the space of analytic functions $A(D)$ on an open interval $D\subset \mathbb R$.

2. Does the function we get by applying this transformation to the exponential function, i.e. $\frac{e^x-1}{x}$ have a name in the literature?

The latter appears in the integrand of the exponential integral; is apparently related to the derivative of the exponential map from Lie groups and occurs in the solution of inhomogeneous linear ODEs, e.g. $\dot x = a x+b, x(t_0)=x_0$ has the solution $ x^*(t) = e^{a(t-t_0)}x_0 + \frac{e^{a(t-t_0)} -1}{a} b$; and in the multivariate case $\dot x(t) = A \cdot x(t) + b , x(t_0) = x_0$ we get the analogous $ x^*(t) = e^{A(t-t_0)}x_0 + \frac{e^{A(t-t_0)}-I}{A}b$. Note that $\frac{e^{A(t-t_0)}-I}{A}$ exists even when $A$ is singular, hence it would be useful to drop this notation and give the function a name instead.

Answer 1

I found the answer I was looking for. It appears that these are known as the $\varphi$-functions in numerical mathematics and they appear in the context of exponential integrators:

• $\varphi_0(z) = e^{z}$
• $\varphi_1(z)=\frac{e^{z}-1}{z}$
• $\varphi_1(z)=\frac{e^{z}-1 -z}{z^2}$
• $\ldots$
• $z\varphi_{n+1}(z) = \varphi_{n}(z)-\frac{1}{(z-1)!}$
• $\varphi_{S}(z)=\frac{1}{(s-1) !} \int_{0}^{1} e^{(1-\theta) z} \theta^{s-1} d \theta$

And they can be computed efficiently using Saad's Trick:

$$\exp\left(\begin{bmatrix} z & 1 \\ 0 & 0 \end{bmatrix}\right) = \begin{bmatrix} \varphi_0(z) & \varphi_1(z) \\ 0 & 1\end{bmatrix}$$

and in particular

$$\exp\left(\begin{bmatrix} A & b \\ 0 & 0 \end{bmatrix}\right) = \begin{bmatrix} \varphi_0(A) & \varphi_1(A)b \\ 0 & 1\end{bmatrix}$$

References:

• Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators, Al-Mohy and Higham (SIAM 2011)
• Matrix Exponential Based Integration Algorithms for Circuit Transient Simulation