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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.

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

Read more about $q$-analogs in this Wikipedia article.

133 questions
87
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1 answer

Conjectured formula for the Fabius function

The Fabius function is the unique function ${\bf F}:\mathbb R\to[-1, 1]$ satisfying the following conditions: a functional–integral equation$\require{action} \require{enclose}{^{\texttip{\dagger}{a poet or philosopher could say "it knows and…
Vladimir Reshetnikov
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27
votes
0 answers

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 coefficient) is defined as $$\binom{n}{k}_q :=…
Henry
  • 3,206
15
votes
0 answers

Different notions of q-numbers

It seems that most of the literature dealing with q-analogs defines q-numbers according to $$[n]_q\equiv \frac{q^n-1}{q-1}.$$ Even Mathematica uses this definition: with the built-in function QGamma you obtain QGamma[n+1,q] / QGamma[n,q] = (q^n-1) /…
André
  • 293
13
votes
4 answers

Intriguing polynomials coming from a combinatorial physics problem

For real $00 $ and integer $k\ge 0$, define $$[k, n]_q \equiv -\sum_{m=1}^{n} q^{m(k+1)} (q^{-n}; q)_m = -\sum_{m=1}^{n} q^{m(k+1)} \prod_{l=0}^{m-1} (1-q^{l-n})$$ where $(\cdot\; ; q)_n$ is a $q$-Pochhammer symbol. These…
Slava Kashcheyevs
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13
votes
1 answer

Which families of groups have interesting formulas for the number of elements of given order?

Suppose that $G$ is a group and that $n$ is a positive integer diving the order of $G$. Let $f_n(G)$ be the number of elements satisfying $x^n = 1$ in $G$. According to a theorem of Frobenius, then we have $f_n(G) \equiv 0 \mod{n}$. Hence if we have…
Mikko Korhonen
  • 26,156
13
votes
2 answers

The groupoid cardinality of $\mathbb{F}_q$-vector spaces $\sum_{n=0}^{\infty} \, \bigl(\prod_{i=0}^{n-1} q^n-q^i\bigr)^{-1}$

Let $q > 1$. What can we say about the value of $$\sum_{n=0}^{\infty} \, \bigl(\prod\limits_{i=0}^{n-1} q^n-q^i\bigr)^{-1} ~~?$$ The series clearly converges. Is there a closed form or something like that? Background: If $q$ is a prime power, then…
Martin Brandenburg
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12
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1 answer

Bijection for $q$-binomial coefficient

Define the $q$-binomial (Gaussian) coefficient ${n+m\brack n}_q$ as the generating function for integer partitions (whose Ferrers diagrams are) fitting into a rectangle $n\times m$, i.e., for the set $P_{n,m}$ of partitions with at most $n$ parts,…
nejimban
  • 4,107
11
votes
2 answers

$q$-analog of Number Theory

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 generalization in some way whilst simultaneously…
Mako
  • 718
11
votes
1 answer

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), 303-306) $C_n(q)=\frac{1}{[n+1]_q}{2n\brack…
J. Yomcha
  • 123
11
votes
0 answers

Irreducibility of q-factorial plus 1

Is it true that $[n]_q! + 1$ is an irreducible polynomial over $\mathbb{Z}$ for all positive integers $n$ ? I checked that this is true for $n$ up to $20$. Here $[n]_q! := 1 (1 + q) (1 + q + q^2) \cdots (1 + q + \cdots + q^{n-1})$ is the…
Penchez
  • 211
10
votes
2 answers

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 binomial coefficient. I have noticed that the…
Bach
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10
votes
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What is the inverse of the $q$-exponential?

The $q$-exponential function is given by the power series $$ e_q(x) = \sum_{n=0}^\infty \frac{x^n}{[n]!} $$ using the $q$-integers $[k]:=q^{k-1}+\cdots+q+1=(q^k-1)/(q-1)$ and $q$-factorials $[n]!=[n][n-1]\cdots[2][1]$. What is the inverse function…
anon
  • 155,259
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
9
votes
4 answers

Proving q-binomial identities

I was wondering if anyone could show me how to prove q-binomial identities? I do not have a single example in my notes, and I can't seem to find any online. For example, consider: ${a + 1 + b \brack b}_q = \sum\limits_{j=0}^{b} q^{(a+1)(b-j)}{a+j…
Nizbel99
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8
votes
1 answer

What does the $q$-Catalan Numbers count?

I had completed a paper describing the $q$-Catalan numbers, which is the $q$-analog of the Catalan numbers. The $n$-th Catalan numbers can be represented by: $$C_n=\frac{1}{n+1}{2n \choose n}$$ and with the recurrence…
user67258
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