# Quantum calculus

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**Quantum calculus**, sometimes called **calculus without limits**, is equivalent to traditional [infinitesimal calculus](/source/Infinitesimal_calculus) without the notion of [limits](/source/Limit_of_a_function). The two types of calculus in quantum calculus are *q*-calculus and *h*-calculus. The goal of both types is to find "analogs" of mathematical objects, where, after taking a certain [limit](/source/Limit_of_a_function), the original object is returned. In *q*-calculus, the limit as *q* tends to 1 is taken of the [*q*-analog](/source/Q-analog). Likewise, in *h*-calculus, the limit as h tends to 0 is taken of the *h*-analog. The parameters q and h can be related by the formula q = e^h.

## Differentiation

The *q*-differential and *h*-differential are defined as:

- d_q(f(x)) = f(qx) - f(x)

and

- d_h(f(x)) = f(x + h) - f(x),

respectively. The [*q*-derivative](/source/Q-derivative) and *h*-derivative are then defined as

- D_q(f(x)) = \frac{d_q(f(x))}{d_q(x)} = \frac{f(qx) - f(x)}{qx - x}

and

- D_h(f(x)) = \frac{d_h(f(x))}{d_h(x)} = \frac{f(x + h) - f(x)}{h}

respectively. By taking the [limit](/source/Limit_of_a_function) as q \rightarrow 1 of the *q*-derivative or as h \rightarrow 0 of the *h*-derivative, one can obtain the [derivative](/source/Derivative):

- \lim_{q \rightarrow 1} D_{q}f(x) = \lim_{h \rightarrow 0}D_h f(x) = \frac{d}{dx}\Bigl( f(x) \Bigr)

## Integration

### q-integral

A function *F*(*x*) is a q-antiderivative of *f*(*x*) if *D*q*F*(*x*) = *f*(*x*). The q-antiderivative (or q-integral) is denoted by \int f(x) \, d_qx and an expression for *F*(*x*) can be found from:\int f(x) \, d_qx = (1-q) \sum_{j=0}^\infty xq^j f(xq^j), which is called the [Jackson integral](/source/Jackson_integral) of *f*(*x*). For 0 < *q* < 1, the series converges to a function *F*(*x*) on an interval (0,*A*] if |*f*(*x*)*x**α*| is bounded on the interval for some 0 ≤ *α* < 1.

The q-integral is a [Riemann–Stieltjes integral](/source/Riemann%E2%80%93Stieltjes_integral) with respect to a [step function](/source/Step_function) having infinitely many points of increase at the points *q**j.*.The jump at the point *q**j* is *q**j*. Calling this step function *g**q*(*t*) gives *dg**q*(*t*) = *d**q**t*.[1]

### h-integral

A function *F*(*x*) is an h-antiderivative of *f*(*x*) if *D**h**F*(*x*) = *f*(*x*). The h-integral is denoted by \int f(x) \, d_hx. If *a* and *b* differ by an integer multiple of *h* then the definite integral \int_a^b f(x) \, d_hx is given by a [Riemann sum](/source/Riemann_sum) of *f*(*x*) on the interval , partitioned into sub-intervals of equal width *h*. The motivation of h-integral comes from the Riemann sum of f(x). Following the idea of the motivation of classical integrals, some of the properties of classical integrals hold in h-integral. This notion has broad applications in [numerical analysis](/source/Numerical_analysis), and especially [finite difference calculus](/source/Finite_difference).

## Example

In infinitesimal calculus, the derivative of the function x^n is nx^{n-1} (for some positive integer n). The corresponding expressions in *q*-calculus and *h*-calculus are:

- D_q(x^n) = \frac{1- q^n}{1 - q} x^{n - 1} = [n]_q\ x^{n - 1}

where [n]_q is the [*q*-bracket](/source/Q-bracket)

- [n]_q = \frac{1 - q^n}{1 - q}

and

- \begin{align} D_h(x^n) &= \frac{(x+h)^n - x^n}{h} \\ &= \frac{1}{h}\left(\sum_{k=0}^n{\binom{n}{k} x^{n-k} h^k - x^n} \right) \\ &= \frac{1}{h}\sum_{k=1}^n{\binom{n}{k} x^{n-k} h^k} \\ &= \sum_{k=1}^n{\binom{n}{k} x^{n-k} h^{k-1}} \\ &= n x^{n - 1} + \frac{n(n-1)}{2} h x^{n - 2} + \cdots + n h^{n-2}x + h^{n - 1}, \end{align}

respectively. The expression [n]_q x^{n - 1} is then the [*q*-analog](/source/Q-analog) and \sum_{k=1}^n{\binom{n}{k} x^{n-k} h^{k-1}} is the *h*-analog of the power rule for positive integral powers. The [q-Taylor expansion](/source/Q-derivative) allows for the definition of *q*-analogs of all of the usual functions, such as the [sine](/source/Sine) function, whose *q*-derivative is the *q*-analog of [cosine](/source/Cosine).

## History

The *h*-calculus is the [calculus of finite differences](/source/Calculus_of_finite_differences), which was studied by [George Boole](/source/George_Boole) and others, and has proven useful in [combinatorics](/source/Combinatorics) and [fluid mechanics](/source/Fluid_mechanics). In a sense, *q*-calculus dates back to [Leonhard Euler](/source/Leonhard_Euler) and [Carl Gustav Jacobi](/source/Carl_Gustav_Jacobi), but has only recently begun to find usefulness in [quantum mechanics](/source/Quantum_mechanics), given its intimate connection with commutativity relations and [Lie algebras](/source/Lie_algebra), specifically [quantum groups](/source/Quantum_group).

## See also

- [Noncommutative geometry](/source/Noncommutative_geometry)
- [Quantum differential calculus](/source/Quantum_differential_calculus)
- [Time scale calculus](/source/Time_scale_calculus)
- [q-analog](/source/Q-analog)
- [Basic hypergeometric series](/source/Basic_hypergeometric_series)
- [Quantum dilogarithm](/source/Quantum_dilogarithm)

## References

1. Abreu, Luis Daniel (2006). ["Functions q-Orthogonal with Respect to Their Own Zeros"](http://www.mat.uc.pt/preprints/ps/p0432.pdf). *Proceedings of the American Mathematical Society*. **134** (9): 2695–2702. [doi:10.1090/S0002-9939-06-08285-2](https://doi.org/10.1090/S0002-9939-06-08285-2). [JSTOR 4098119](https://www.jstor.org/stable/4098119)

## Further reading

- George Gasper, Mizan Rahman, *Basic Hypergeometric Series*, 2nd ed, Cambridge University Press (2004), ISBN 978-0-511-52625-1, [doi:10.1017/CBO9780511526251](https://doi.org/10.1017/CBO9780511526251)

- Jackson, F. H. (1908). "On *q*-functions and a certain difference operator". *Transactions of the Royal Society of Edinburgh*. **46** (2): 253–281. [doi:10.1017/S0080456800002751](https://doi.org/10.1017/S0080456800002751). [S2CID 123927312](https://api.semanticscholar.org/CorpusID:123927312)
- Exton, H. (1983). *q-Hypergeometric Functions and Applications*. New York: Halstead Press. ISBN 0-85312-491-4.
- Kac, Victor & Cheung, Pokman (2002). *Quantum calculus*. Universitext. Springer-Verlag. ISBN 0-387-95341-8.

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