# Tagged Questions

The tag has no usage guidance.

378 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 ...
1k 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 ...
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 $$\frac{1}{(q;q^2)_\infty}\sum \limits_{j =0}^... 1answer 605 views ### Are the following q-Genocchi numbers known? The sequence of Genocchi numbers {({G_{2n}})_{n \ge 0}}= (0,1,1,3,17,155,2073,...) can be defined by the generating function z\frac{{1 - {e^z}}}{{1 + {e^z}}} = \sum {{{( - 1)}^n}{G_{2n}}\frac{... 0answers 82 views ### notation for (a-b)(a-qb)\dots (a-q^{n-1}b) I wonder whether there is a notation for such thing, which I denote [a;b]_q^n for a moment:$$ [a;b]_q^n:=(a-b)(a-qb)\dots (a-q^{n-1}b)=a^n(b/a;q)_n, $$this last equation uses q-Pochhammer symbol ... 0answers 73 views ### Identities for {~}_3\phi_1? I am looking for some source of summation formulas for the q-hypergeometric function {~}_3\phi_1 in the sense of Gasper-Rahman book. For some reason, the book focuses heavily on {~}_{r+1}\phi_r ... 1answer 93 views ### Expressions involving q-binomial coefficients? I have bumped into the following expressions involving q-binomial coefficients.$$ \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\...
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 ...