Quantum calculus encompasses $q$-calculus and $h$-calculus, and is a notion of "calculus without limits". Do not confuse with the (quantum-mechanics) tag. For questions on Schrodinger's equation and solutions, use (quantum-mechanices), (pde), (fourier-analysis), and/or (calculus) as appropriate.
Questions tagged [quantum-calculus]
22 questions
9
votes
1 answer
Is there a gamma-like function for the q-factorial?
I'm looking at quantum calculus and just trying to understand what is going with this subject. Looking at the q-factorial made me wonder if this function could take all real or even complex numbers in the same way that $\Gamma (z)$ works as an…
futurebird
- 6,308
3
votes
1 answer
Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence
Consider the density matrices with the following spectral decompositions:
$$\rho=\lambda_1|\nu_1\rangle+\lambda_{2}|\nu_2\rangle$$
and
$$\sigma=\gamma_1|\omega_1\rangle+\gamma_2|\omega_2\rangle$$
such that $\gamma_i=\lambda_i$ and…
James Smithson
- 304
2
votes
2 answers
Source for these $q$-binomial identities
Recently, I found two $q$-binomial identities
$$
\sum_{p=0}^b \binom{a}{p}_q q^{\binom{p}{2}} (-1)^p = \binom{b-a}{b}_q q^{ab},
$$
and
$$
\sum_{p=a}^b \binom{p-1}{a-1}_q q^{-ap} = \binom{b}{a}_q q^{-ab}.
$$
Even though they look very simple, and…
Nolord
- 2,233
2
votes
0 answers
Why is $q$ sometimes a complex number, but other times a prime power?
In the fields of representation theory and quantum algebra, we often start with some $\mathbf{C}$-algebra and study it's quantization as an algebra over $\mathbf{C}(q)$, using the algebra structure to twist the multiplication. But what is $q$?…
Mike Pierce
- 19,406
2
votes
1 answer
Studying quantized algebras, what motivates the choice of base ring?
In the fields of representation theory and quantum algebra, we often start with, for example, some $\mathbf{C}$-algebra $A$ and study a quantization of $A$ by adjoining an indeterminate $q$, or sometimes $v$, to twist the multiplication in $A$. Now,…
Mike Pierce
- 19,406
2
votes
1 answer
Intuition behind this q-binomial formula counting sums of subsets
We have these transparent, motivating interpretations for binomial coefficients and their $q$-analogue.
$$
\binom{n}{j} = \begin{matrix} \text{"The number of subsets of size $j$}\\ \text{of a set of size $n$."} \end{matrix}
\qquad
…
Mike Pierce
- 19,406
2
votes
0 answers
Q-Exponential Sum Identity
We define the "standard" q-exponential as follows
$$ e_q(x) = 1 + \frac{1}{1} x+ \frac{1}{(1+q)} x^2 + \frac{1}{(1+q)(1+q+q^2)}x^3 ... =$$
$$ \sum_{i = 0}^{\infty} \left[ (1-q)^ix^i \prod_{j=1}^{i} \frac{1}{(1-q^j)} \right]$$
What I'm interested…
Sidharth Ghoshal
- 18,359
2
votes
1 answer
double integrals on quantum calculus
I need references or book recommendations to find properties of double integrals on quantum calculus. Especially i need analogue of Fubini's theorem on q-calculus.
Raio
- 1,545
2
votes
0 answers
improper integrals in q-calculus
In quantum calculus is this equality possible for improper integrals?
$\lim_{x\to\infty}\int_0^xf(t)d_qt=\int_0^\infty f(x)d_qx$
Raio
- 1,545
1
vote
1 answer
Ejemplos de la integral de Jackson (Examples of Jackson's Integral)
Original question in Spanish
La integral de Jackson está definida en el cálculo cuántico, y quisiera que alguien me ayudara a la explicación de un ejemplo de este estilo de integrales. Gracias
Added
Translation:
The Jackson integral is defined in…
lizeth
- 11
1
vote
1 answer
Studying quantized algebras, why introduce $q^{1/2}$ instead of just $q$?
In the fields of representation theory and quantum algebra, we often study quantized versions of algebraic objects by regarding them as algebras over $\mathbf{C}(q)$, or some subring of $\mathbf{C}(q)$, and using that algebra structure to twist the…
Mike Pierce
- 19,406
1
vote
1 answer
What is the way to change the limits of q-integration of a double integral
What is the way to change the limits of q-integration of a double integral.
For exemple, what is the answer after change the order of integration of $$\int_0^1 \int_0^{x} f(x,y)\ d_qy\ d_qx$$
https://en.wikipedia.org/wiki/Jackson_integral
Martin Bokner
- 121
1
vote
0 answers
What does the statement "x is a diagram of classical logics" mean?
From the Wikipedia entry on quantum logic
A more modern approach to the structure of quantum logic is to assume that it is a diagram – in the sense of category theory – of classical logics (see David Edwards).
What does the author mean by the…
Hal
- 3,456
0
votes
0 answers
q-differentiability in $R_q$
Wokring in q-calculus where everything is defined on the set $$R_q =\{\pm q^k,k \in \mathbb{Z} \} \cup \{ 0\}$$
In which they define the q-derivative (or q-difference operator) as
$$D_qf(x)=\frac{f(x)−f(qx)}{(1−q)x}, \quad x\neq 0$$
$$D_qf(x)=f'(0),…
0
votes
0 answers
Combinatorics - Q Calculus Pascal's Identity Proof
I have been trying to get started on this simple combinatorics proof. This has led me to start a proof by induction using pascal's identity from https://en.wikipedia.org/wiki/Gaussian_binomial_coefficient#Analogs_of_Pascal's_identity. Not sure…
