Discrete Calculus

A sequence \((a_n)_{n\in \mathbb{N}}\) can be reformulated as a function from natural number (or integer) to a set \(S\). \[ a\colon \mathbb{N} \to S. \]

Discrete calculus mostly deals with a sequence in real or complex number.

Falling and rising factorials are the discrete equivalent of power functions.

\begin{align*} \Delta n^{\underline{p}}&=(n+1)n^{\underline{p-1}} - (n-p+1)n^{\underline{p-1}} \\ &=p\, n^{\underline{p-1}} \end{align*}

Exponential function with base 2 is the discrete equivalent of the natural exponential function \(e^x\). \[ \Delta 2^n = 2^n \]

Linearity applies \[ \Delta(a f_n + b g_n) = a\Delta f_n + b\Delta g_n, \] but Leibniz rule (product rule) does not \[ \Delta (f_ng_n) \\= \Delta f_ng_n + f_{n+1}\Delta g_n = \Delta f_ng_{n+1} + f_n\Delta g_n \\=\Delta f_ng_n + f_n\Delta g_n + \Delta f_n \Delta g_n. \] This is directly related to the summation by parts, and the form is similar to the quadratic variation

Quotient Rules

• \[ \Delta \left(\frac{1}{a_n}\right) = -\frac{\Delta a_n}{a_{n+1}a_n} = -\frac{\Delta a_n}{a_n^2 + a_n\Delta a_n} \]
• \[ \Delta \left(\frac{a_n}{b_n}\right) = \frac{\Delta a_n b_n - a_n\Delta b_n}{b_n(b_n+\Delta b_n)} \]

Higher Order Difference

• \[ \Delta^k a_n = \sum_{i = 0}^k(-1)^i\binom{k}{i}a_{n+k-i} \]

The linear operator \(L\colon a_{n+m} \mapsto a^m\) also maps \[ L\colon \Delta^m a_n \mapsto (a-1)^m. \] See umbral calculus.

\[ a_{n+k} = \sum_{i=0}^k \binom{k}{i}\Delta^ia_n \]

It applies to continuous function as well, \[ f(x) = \sum_{n=0}^\infty \binom{x}{n}\Delta^n f(0). \]

• Abel's Lemma, Abel Transformation

\[ \sum_{k=m}^n f_k\Delta g_k = (f_{n+1}g_{n+1}-f_mg_m)-\sum_{k=m}^n g_{k+1}\Delta f_k \]

\[ \Delta y_n + ay_n = 0 \]

The solution is \(y_n = C(1-a)^n\)

\[ a_ky_{n+k} + a_{k-1}y_{n+k-1} + \cdots + a_1y_{n+1} + a_0 y_n = 0 \]

The solution is the linear combination of the powers of the roots of characteristic equation: \[ a_k\alpha^k + a_{k-1}\alpha^{k-1} + \cdots + a_1\alpha + a_0 = 0. \]

The full solution is given by: \[ y_n = C_1 \alpha_1^n + C_2\alpha_2^n +\cdots + C_k\alpha_k^n \]

The homogeneous solution with the particular solution.

For the constant case: \[ a_ky_{n+k} + a_{k-1}y_{n+k-1} + \cdots + a_1y_{n+1} + a_0 y_n = s \] the particular solution is a constant.

\[ \mathbf{x}_{t+2} = \mathbf{A}\mathbf{x}_{t+1} + \mathbf{B}\mathbf{x}_t \]

Power function \[ \phi(x^n) = \sum_{k=1}^n (-1)^{n-k}\left[\vphantom{1}n \atop k\right]x^k \] \[ \phi^{-1}(x^n) = \sum_{k=1}^n \left\{\vphantom{1}n\atop k\right\}x^k \] where the coefficients are Stirling numbers.

Exponential Function \[ \phi(e^{ax}) = \sum_{n=0}^\infty \binom{x}{n}a^n = (a+1)^x \]

Trigonometric Function \[ \phi(\sin x) = \sqrt{2^x}\sin\left(\frac{\pi x}{4}\right) \]

Logarithmc Function \[ \phi(\ln x) = H_{x-1} \] where \(H_x\) is the .

\[ X_{k}=\sum_{n=0}^{N-1}x_n\cdot e^{-i2\pi k n/N} \] for \(\lbrace x_n\rbrace_{n=0}^{N-1}, \lbrace X_k\rbrace_{k=0}^{N-1}\).

\[ \mathcal{Z}\{x_n\}(z) = \sum_{n=-\infty}^{\infty}x_nz^{-n}. \]

\(z^{-n}\) is used because it is discrete equivalent of exponential function.

See equivalences.