Quantum calculus

In the q-calculus and h-calculus, differentials of functions are defined as

${\displaystyle d_{q}(f(x))=f(qx)-f(x)}$

and

${\displaystyle d_{h}(f(x))=f(x+h)-f(x)}$

respectively. Derivatives of functions are then defined as fractions by the q-derivative

${\displaystyle D_{q}(f(x))={\frac {d_{q}(f(x))}{d_{q}(x)}}={\frac {f(qx)-f(x)}{(q-1)x}}}$

and by

${\displaystyle D_{h}(f(x))={\frac {d_{h}(f(x))}{d_{h}(x)}}={\frac {f(x+h)-f(x)}{h}}}$

In the limit, as h goes to 0, or equivalently as q goes to 1, these expressions take on the form of the derivative of classical calculus.

The derivative of the function ${\displaystyle x^{n}}$ (for some positive integer ${\displaystyle n}$) in the classical calculus is ${\displaystyle nx^{n-1}}$. The corresponding expressions in q-calculus and h-calculus are

${\displaystyle D_{q}(x^{n})={\frac {q^{n}-1}{q-1}}x^{n-1}=[n]_{q}\ x^{n-1}}$

with the q-bracket

${\displaystyle [n]_{q}={\frac {q^{n}-1}{q-1}}}$

and

${\displaystyle D_{h}(x^{n})=nx^{n-1}+{\frac {n(n-1)}{2}}hx^{n-2}+\cdots +h^{n-1}}$

respectively. The expression ${\displaystyle [n]_{q}x^{n-1}}$ is then the q-calculus analogue of the simple power rule for positive integral powers. In this sense, the function ${\displaystyle x^{n}}$ is still nice in the q-calculus, but rather ugly in the h-calculus – the h-calculus analog of ${\displaystyle x^{n}}$ is instead the falling factorial, ${\displaystyle (x)_{n}:=x(x-1)\cdots (x-n+1).}$ One may proceed further and develop, for example, equivalent notions of Taylor expansion, et cetera, and even arrive at q-calculus analogues for all of the usual functions one would want to have, such as an analogue for the sine function whose q-derivative is the appropriate analogue for the cosine.

The h-calculus is just the calculus of finite differences, which had been studied by George Boole and others, and has proven useful in a number of fields, among them combinatorics and fluid mechanics. The q-calculus, while dating in a sense back to Leonhard Euler and Carl Gustav Jacobi, is only recently beginning to see more usefulness in quantum mechanics, having an intimate connection with commutativity relations and Lie algebra.

• Noncommutative geometry
• Quantum differential calculus
• Time scale calculus
• q-analog
• Basic hypergeometric series
• Quantum dilogarithm

• George Gasper, Mizan Rahman, Basic Hypergeometric Series, 2nd ed, Cambridge University Press (2004), ISBN 978-0-511-52625-1, doi: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. S2CID 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.