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This wiki is a resource for people who do research in time scale calculus. Time scale calculus is a unification and extension of differential and difference calculus in which one does calculus upon a set $\mathbb{T}$ of real numbers called a time scale. When $\mathbb{T}=\mathbb{R}$ the resulting theory becomes differential calculus, when $\mathbb{T}=\mathbb{Z}$ the resulting theory becomes difference calculus, and when $\mathbb{T}=\{1,q,q^2,\ldots\}, q>1$, the resulting theory becomes the $q$-calculus. Time scales also include any closed subset of $\mathbb{R}$, so more exotic sets such as the Cantor set are also subsumed in the theory.

A result proven in time scale calculus implies the result for all choices of $\mathbb{T}$ so a result in time scale calculus immediately implies the result in differential calculus, the same result in difference calculus, the same result in $q$-calculus, the same result in calculus on the Cantor set, and countless others. For an example of this phenomenon, see the familiar properties of the $\Delta$-derivative to classical differentiation or to taking a forward difference.

See the Python library timescalecalculus on GitHub and its documentation.

Registration Due to a resurgence of automated spam bots, account registration and anonymous editing is currently disabled. Please contact Tom Cuchta (tomcuchta@gmail.com) to gain access to edit the wiki.

Time scales calculus

Examples of time scales
$\Huge\mathbb{R}$ Real numbers	$\Huge\mathbb{Z}$ Integers	$\Huge{h\mathbb{Z}}$ Multiples of integers	$\Huge\mathbb{Z}^2$ Square integers	$\Huge\mathbb{H}$ Harmonic numbers	$\Huge\mathbb{T}_{\mathrm{iso}}$ Isolated points
$\Huge\sqrt[n]{\mathbb{N}_0}$ nth root numbers	$\Huge\mathbb{P}_{a,b}$ Evenly spaced intervals	$\huge\overline{q^{\mathbb{Z}}}$ Quantum, $q>1$	$\huge\overline{q^{\mathbb{Z}}}$ Quantum, $q<1$	$\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$ Closure of unit fractions	$\Huge\mathcal{C}$ Cantor set

$\Delta$-special functions on time scales
$\cos_p$	$\cosh_p$	$e_p$	$g_k$	$h_k$	$\sin_p$	$\sinh_p$

Hilger complex plane and friends
$\Huge\mathbb{A}_h$ Hilger alternating axis	$\Huge\mathbb{I}_h$ Hilger circle	$\Huge\mathbb{C}_h$ Hilger complex plane	$\Huge\mathrm{Im}_h$ Hilger imaginary part	$\Huge\mathring{\iota}$ Hilger pure imaginary	$\Huge\mathbb{R}_h$ Hilger real axis	$\Huge\mathrm{Re}_h$ Hilger real part

Bilateral Laplace transform
Unilateral Laplace transform
Cauchy function
Calculus of variations
Chain rule
Unilateral convolution
Dense point
Disconjugate
Dynamic equation
Forward circle minus
Backward circle minus
Forward circle plus
Backward circle plus
Complex calculus on time scales
Convergence of time scales
Dilation of time scales
Duality of $\Delta$ and $\nabla$
Fractional calculus
Frequency roots
Generalized square
Generalized zero
Induction on time scales
L'Hospital's Rule
First mean value theorem
Pre-differentiable
Marks-Gravagne-Davis Fourier transform
Cuchta-Georgiev Fourier transform
rd-continuous
Forward regressive function
Regulated function
Riccati equation
Riesz representation theorem
Scattered point
Self-adjoint
Shifting problem
Substitution
Variation of parameters
Wronskian

$\Delta$-calculus

Completely delta differentiable
$\Delta$-derivative
$\Delta$ heat equation
$\Delta$-integral
$\Delta$-Taylor's formula
$\Delta$ wave equation
Directional $\Delta$ derivative
Partial $\Delta$ derivative
Partial $\Delta$ dynamic equations

$\nabla$-calculus

$\nabla$-derivative
$\nabla$-integral

$\Diamond_{\alpha}$-calculus

$\Diamond_{\alpha}$-derivative
$\Diamond_{\alpha}$-Hölder inequality
$\Diamond_{\alpha}$-Jensen's inequality
$\Diamond_{\alpha}$-Minkowski's inequality
$\Diamond$-integral

Probability Theory

• Cumulant generating function
• Cumulative distribution function
• Moments
• Probability density function
• Joint time scales probability density function
• Moment generating function
• Expected value
• Variance

Examples of time scales

1. The real line: $\mathbb{R}$
2. The integers: $\mathbb{Z} = \{\ldots, -1,0,1,\ldots\}$
3. Multiples of integers: $h\mathbb{Z} = \{ht \colon t \in \mathbb{Z}\}$
4. Quantum numbers ($q>1$): $\overline{q^{\mathbb{Z}}}$
5. Quantum numbers ($q<1$): $\overline{q^{\mathbb{Z}}}$
6. Square integers: $\mathbb{Z}^2 = \{t^2 \colon t \in \mathbb{Z} \}$
7. Harmonic numbers: $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$
8. The closure of the unit fractions: $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
9. Isolated points: $\mathbb{T}=\{\ldots, t_{-1}, t_{0}, t_1, \ldots\}$

Special functions on time scales

$\mathrm{c}_{pq}$
$\mathrm{ch}_{pq}$
$\mathrm{s}_{pq}$
$\mathrm{sh}_{pq}$
Gamma function
Hyperbolic functions
Euler-Cauchy logarithm
Bohner logarithm
Jackson logarithm
Mozyrska-Torres logarithm
Gaussian bell

$\nabla$-calculus

$\nabla \widehat{\cos}_p$
$\nabla \widehat{\cosh}_p$
$\nabla \widehat{\exp}$
$\nabla \hat{h}_k$
$\nabla \hat{g}_k$
$\nabla \widehat{\sin}_p$
$\nabla \widehat{\sinh}_p$

$\Diamond_{\alpha}$-calculus

$\Diamond_{\alpha}$-$\cos_p$
$\Diamond_{\alpha}$-$\cosh_p$
$\Diamond_{\alpha}$-$e_p$
$\Diamond$-$\sin_p$
$\Diamond$-$\sinh_p$

Probability Distributions on time scales

Uniform distribution
Exponential distribution
Gamma distribution

Special cases of time scales

Differential equations

Hypergeometric differential equation
Confluent hypergeometric differential equation

Difference equations

Difference equation of hypergeometric type

$q$-difference equations

q-difference equation of hypergeometric type