# Tagged Questions

118 views

### a question on sum of Gaussian binomial coefficients

I was trying to calculate something and at some point I get the following sum: \sum_{t=0,t \text{ even}}^{s}{s+3n \brack s-t}\sum_{i = 0}^{t/2}q^{2i^2}{t/2+2n-i \brack t/2-i}{n ...
75 views

### Quadratic transformation of hypergeometric function 2F1

I want to know whether there is some transformation between $_2F_1(a,b;c;x)$ and $_2F_1(a',b';c';x(1-x))$. Here is an example called the Kummer quadratic transformation, which may be known to most of ...
199 views

Let $m,k$ be positive integers with $k\le m$. Does anyone know some hypergeometric identities that imply $$\sum_{j=0}^k\frac{(-1/2)_{k-j}(m+1)_j(-m)_j}{(1/2)_j(k-j)!j!} ... 0answers 302 views ### Polynomials with presumably positive coefficients The q-Pochhammer symbol (q) _ 0:=1 and (q) _ n:=\prod _ {j=1}^n (1-q^j) for n > 0 is clearly a polynomial in q which has both positive and negative coefficients when n>0. The ... 4answers 2k views ### Integer-valued factorial ratios This historical question recalls Pafnuty Chebyshev's estimates for the prime distribution function. In his derivation Chebyshev used the factorial ratio sequence$$ ...
One of the recent questions, in fact the answer to it, reminded me about the binomial sequence $$a_n=\sum_{k=0}^n{\binom{n}{k}}^2{\binom{n+k}{k}}^2, \qquad n=0,1,2,\dots,$$ of the Apéry ...