# Questions tagged [q-analogs]

Use this tag for questions pretaining to q-analogs of functions, for example q-Binomials, $q$-derivatives, the q-theta function, the q-Pochhammer symbol, etc.

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• 31
1 vote
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### $q$-derivative satisfying linear approximation characterization

Suppose instead of defining the derivative as $$f'(x) = \lim_{h\to 0}\frac{f(x + h) - f(x)}{h}$$ we define it as $$f_q'(x) = \lim_{q\to 1}\frac{f(qx) - f(x)}{qx - x}.$$ Does this alternative ...
• 369
1 vote
106 views
+50

### $q$-Pochhammer at root of unity

Are there any identities, papers/studies, posts, etc that go over $$(\ln\zeta_n^k;q)_{\infty} = \prod_{m=0}^{\infty}(1-\frac{2\pi i k q^m}{n})$$ which is sometimes called the $q$-Pochhammer or quantum ...
• 690
85 views

### Proving Gauss’s summation theorem for the $q$-binomial coefficients

I am following Warren P. Johnson's "An Introduction to q-analysis". We are supposed to prove $$\binom{n+1}{k+1}_q=\sum_{m=k}^{n}q^{m-k}\binom{m}{k}_q$$ Here is my attempt. We start off by ...
• 23
70 views

### $(p,q)$-Weyl Algebra

In this Introduction to representation theory they define the '$q$-Weyl algebra by the primary defining relation $$xy = qyx$$ This seems appropriate in $q$-deformations based on the basic building ...
• 690
1 vote
30 views

### ${[n]}_{q,q^{-1}}$ $q$-deformation

It seems that in some $q$-deformations the following definition of a $q$-number is used: $$(n)_q = \frac{q^n-q^{-n}}{q-q^{-1}}$$ If we define $${[n]}_q = \frac{1-q^n}{1-q}$$ as the 'conventional' $q$-...
• 690
101 views

### Discriminant of numbers

In a previous post on a $q$-analog of number theory I present the fact that any $q$-number can be written as a unique product of cyclotomic polynomials, similar to the fundamental theorem of ...
• 690
1 vote
75 views

### Looking for a proper name for this vector subspaces

Let $K^n$ be the vector space of rows of length $n$ over $K$. A subspace $S$ is called coordinately complete if for every $i\ (1 \leq i \leq n)$ there is an element in $S$ with a nonzero $i$th ...
422 views

### $q$-analog of Number Theory [closed]

The main motivation behind this is to see whether the 'magic' of q-analogs can be felt in number theory. Obviously for q-analogs to be applied to number theory the parametrization in $q$ must yield a ...
• 690
63 views

• 504
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### Search for an algebraic proof of a near q-Vandermonde identity.

When trying to prove this q-binomial identity I had soon the idea that here we have a q-Vandermonde identity in disguise. I could transform the identity to \begin{align*} \color{blue}{(-1)^mq^{\binom{...
• 109k
305 views

### Prove $\sum_{i=0}^{n}(-1)^{i}q^{(i+1)i/2}{n\choose i}_{q}{n+m-i\choose n}_{q}=1$

Show that for any non-negative integers $n, m$ such that $n\le m$, we have $$\sum_{i=0}^{n}(-1)^{i}q^{(i+1)i/2}{n\choose i}_{q}{n+m-i\choose n}_{q}=1$$ where ${n\choose i}_{q}$ is the Gaussian ...
• 5,750
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### Is there a $q$-analog for the product of binomial coefficients?

The $q$-analog of the binomial coefficient $\binom{n}{k}$ may be defined as the coefficient of $x^k$ in $\prod_{i=0}^{n-1}(1+q^ix)$. Classical arithmetic identities tend to have $q$-analogs. I am ...
• 1,881
49 views

### q analogue of a number is a polynomial in $[2]_q$

$[m]_q = \frac{q^m−q^{-m}}{q-q^{-1}}$ is the q- analogue of the number $m\in\mathbb{Z}_{\geq0}$. $[0]_q=0$, $[1]_q=1$ and $[2]_q=q+q^{-1}$. I don't know how to prove Every $[m]_q$ can be expressed as ...
• 369
67 views

I am trying to prove the following Given $n \in \mathbb{N}$ we define $[n]_{q} = (1-q^{n})/(1-q)$. We also define $[n]_{q} ! = [n-1]_{q} ! \cdot [n]_{q}$, with $[1]_{q} ! =1$. Then I want to prove the ...
• 561
960 views

### A curious identity on $q$-binomial coefficients

Let's first recall some notations: The $q$-Pochhammer symbol is defined as $$(x)_n = (x;q)_n := \prod_{0\leq l\leq n-1}(1-q^l x).$$ The $q$-binomial coefficient (also known as the Gaussian binomial ...
• 3,136
69 views

### Common divisor of Gaussian coefficient expressions

I have a question about common divisors of some expressions involving Gaussian coefficients, in particular in the case ${n \brack 1}_{q} = \frac{q^{n}-1}{q-1}$ where $q$ is a prime power. It is well ...
• 13.4k
78 views

• 31
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• 137
52 views

### Divisor Polynomials

For an integer $n>0$, we may define a polynomial $$p_n(x) = \sum_{d\mid n} x^d.$$Have such polynomials been seriously studied and do they have any interesting properties? Looking at this more, I ...
• 1,828
194 views

### Recurrence formula of the MacMahon $q$-analog of the Catalan numbers

Catalan number is defined by $C_{n}=\frac{1}{n+1}\binom{2n}{n}.$ Two natural $q$-analogs of Catalan numbers are (see Carlitz and Scoville, A note on weighted sequences, Fibonacci Quarterly, 13 (1975), ...
• 123
1 vote