Questions tagged [q-identities]

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6
votes
1answer
217 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
2answers
627 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}...
9
votes
0answers
159 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 ...
3
votes
1answer
522 views

On Ramanujan's beautiful cubic identity

Let $a_i, b_i, c_i$ be defined by the following$\colon$ $\frac{1 + 53X + 9X^2}{1 - 82X - 82X^2 + X^3} = a_0 + a_1X + \ldots$. $\frac{2 - 26X - 12X^2}{1 - 82X - 82X^2 + X^3} = b_0 + b_1X + \ldots$. ...
8
votes
1answer
251 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 ...
5
votes
1answer
144 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}^...
8
votes
1answer
508 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 ...
4
votes
0answers
81 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 ${~}_{r+1}\phi_r$ ...
2
votes
1answer
111 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
0answers
253 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 ...
6
votes
2answers
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 ...
4
votes
1answer
637 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{...