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7
votes
1answer
342 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
59 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
76 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 ...
4
votes
2answers
816 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 ...
3
votes
1answer
573 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
vote
0answers
212 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 ...