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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 but when $\mathbb{T}=\mathbb{Z}$ the resulting theory becomes difference 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.

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Calculus on time scales

Examples of time scales

Real numbers
Multiples of integers
Square integers
Harmonic numbers
Isolated points
nth root numbers
Evenly spaced intervals
Quantum, $q>1$
Quantum, $q<1$
$\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
Closure of unit fractions
Cantor set
$\Delta$-special functions on time scales








Abel's theorem
Bilateral Laplace transform
Cauchy function
Calculus of variations
Chain rule
Dense point
Dynamic equation
Circle minus
Circle plus
Complex calculus on time scales
Convergence of time scales
Dilation of time scales
Duality of $\Delta$ and $\nabla$
Fractional calculus
Frequency roots
Function spaces
Generalized square
Generalized zero
Hilger alternating axis
Hilger circle
Hilger complex plane
Hilger imaginary part
Hilger pure imaginary
Hilger real axis
Hilger real part
Induction on time scales
Laplace transform
L'Hospital's Rule
Mean value theorem
Fourier transform
Regressive function
Regulated function
Riccati equation
Riesz representation theorem
Scattered point
Shifting problem
Variation of parameters


Completely delta differentiable
$\Delta$-Bernoulli inequality
$\Delta$-Bihari inequality
$\Delta$-Cauchy-Schwarz inequality
$\Delta$-Gronwall inequality
$\Delta$ heat equation
$\Delta$-Hölder inequality
$\Delta$-Jensen inequality
$\Delta$-Lyapunov inequality
$\Delta$-Markov inequality
$\Delta$-Minkowski inequality
$\Delta$-Opial inequality
$\Delta$-Taylor's formula
$\Delta$-Tschebycheff inequality
$\Delta$-Wirtinger inequality
$\Delta$ wave equation
Directional $\Delta$ derivative
Partial $\Delta$ derivative
Partial $\Delta$ dynamic equations




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

Probability Theory

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

Gamma function
Hyperbolic functions
Euler-Cauchy logarithm
Bohner logarithm
Jackson logarithm
Mozyrska-Torres logarithm
Gaussian bell


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



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