Questions tagged [q-identities]
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16
questions
12
votes
0
answers
515
views
$q$-analogue of the multinomial theorem?
The $q$-binomial theorem states that
$$
\prod_{k=0}^{n-1}(1+q^kt) = \sum_{k=0}^n q^{\binom k2}{n\brack k}_q t^k.
$$
This identity is a $q$-analogue of the binomial theorem
$$
(1+t)^n = \sum_{k=0}^n \...
11
votes
1
answer
582
views
Quest for a human proof of a $q-$binomial identity
Let $$f(n,k) = \sum\limits_{j = - k}^k {{{( - 1)}^{k - j}}}
\binom{n-j}{k-j}\binom{n+j}{k+j}.$$
Then $f(n,k)=\binom{n}{k}$
because it satisfies $f(n,k)=f(n-1,k)+f(n-1,k-1)$ and the obvious ...
9
votes
1
answer
401
views
notation for $(a-b)(a-qb)\dots (a-q^{n-1}b)$
I wonder whether there is a notation for such thing, which I denote $[a;b]_q^n$ for a moment:
$$
[a;b]_q^n:=(a-b)(a-qb)\dots (a-q^{n-1}b)=a^n(b/a;q)_n,
$$
this last equation uses $q$-Pochhammer symbol ...
9
votes
0
answers
191
views
For $q$-analogues of a known curious identity
In 2002 I published the folllowing curious combinatorial identity:
$$(x+m+1)\sum_{i=0}^m(-1)^i\binom{x+y+i}{m-i}\binom{y+2i}i-\sum_{i=0}^m\binom{x+i}{m-i}(-4)^i=(x-m)\binom xm.$$
My original proof is ...
7
votes
2
answers
2k
views
series expansion of the q-Pochhammer symbol
The following identity arose while I was working on a recent MO question:
$-\sum_{n=1}^{\infty}\frac{1}{n}\frac{(-x)^n}{1-x^n}=\sum_{n=1}^{\infty}\frac{1}{n}\frac{x^n}{1-x^{2n}}.$
I have no doubt ...
7
votes
1
answer
303
views
A curious $q$-series identity on a truncated Euler function
Recall that a $q$-Pochhammer symbol is defined as
$$
(x)_n = (x;q)_n := \prod_{l=0}^{n-1}(1-q^l x).
$$
I found the following curious $q$-series identity that seems to hold for any $n\geq 0$:
$$
(-1)^{...
6
votes
1
answer
256
views
A $q$-series identity: proof request
Denote the $q$-expressions $[q]_n=(1-q)\cdots(1-q^n)$ for $n\geq1$ and $[q]_0:=1$. Also, $[q]_{\infty}=(1-q)(1-q^2)(1-q^3)\cdots$.
QUESTION. Is this identity true? It seems to be.
$$\sum_{n=0}^{\...
5
votes
2
answers
861
views
Searching for a proof for a series identity
The below identity I have found experimentally.
Question. Is this true? If so, may you provide a "slick" (or any) proof.
$$6\sum_{k=1}^{\infty}\frac{k^2q^k}{(1-q^k)^2}+12\left(\sum_{k=1}^{\infty}...
5
votes
1
answer
184
views
Reference request for simple $q-$ identities
I stumbled upon the following simple $q-$identities:
$$\frac{1}{(-q;q)_\infty}\sum \limits_{j =0}^{\infty}\frac{q^{(2r+1)j}}{(q^2;q^2)_j}=(q;q^2)_r$$
and
$$\frac{1}{(q;q^2)_\infty}\sum \limits_{j =0}^...
4
votes
2
answers
236
views
(Conceptual) proof and/or interpretation of a $q$-binomial identity
There is a $q$-binomial identity that I encountered in one paper I am reading (https://arxiv.org/abs/1910.06193) which probably admits a very simple proof that I do not see: for two nonnegative ...
4
votes
1
answer
643
views
On Ramanujan's beautiful cubic identity
Let $a_i, b_i, c_i$ be defined by the following$\colon$
$\frac{1 + 53X + 9X^2}{1 - 82X - 82X^2 + X^3} = a_0 + a_1X + \ldots$.
$\frac{2 - 26X - 12X^2}{1 - 82X - 82X^2 + X^3} = b_0 + b_1X + \ldots$.
...
4
votes
1
answer
665
views
Are the following q-Genocchi numbers known?
The sequence of Genocchi numbers
${({G_{2n}})_{n \ge 0}}=$ $(0,1,1,3,17,155,2073,...)$
can be defined by the generating function
$z\frac{{1 - {e^z}}}{{1 + {e^z}}} = \sum {{{( - 1)}^n}{G_{2n}}\frac{...
4
votes
0
answers
89
views
Identities for ${~}_3\phi_1$?
I am looking for some source of summation formulas for the $q$-hypergeometric function ${~}_3\phi_1$ in the sense of Gasper-Rahman book. For some reason, the book focuses heavily on ${~}_{r+1}\phi_r$ ...
3
votes
1
answer
562
views
Sum of $q$-binomial coefficients
Denote by $ \binom{n}{k}_q = \prod_{i=0}^{k-1} \frac{ q^{n-i} - 1 }{ q^{k-i} - 1 } $, $ k = 0, 1, \ldots, n $, the $ q $-binomial (Gaussian) coefficients. These numbers are symmetric, in the sense ...
2
votes
1
answer
129
views
Expressions involving $q$-binomial coefficients?
I have bumped into the following expressions involving $q$-binomial coefficients.
$$
\sum_{s=0}^a (-1)^s q^{s^2-s} \left(\begin{array}{c}2b+1-2s\\2a-2s\end{array}\right)_q
\left(\begin{array}{c}b\\s\...
1
vote
0
answers
269
views
references for q-series identities
I need references for
$\sum_{n=0}^N\frac{q^n}{(q^2;q^2)_n(q^2;q^2)_{N-n}}=\frac{(-q,q)_N}{(q^2;q^2)_N}$
and
$\sum_{n=0}^N\frac{(-1)^nq^{n^2}}{(q^2;q^2)_n(q;q)_{N-n}}=\frac1{(q^2;q^2)_N}$
A ...