# Tagged Questions

**2**

votes

**2**answers

751 views

### Proving generating functions equality

What do you use to prove the following equality (and possibly more general ones of the kind)?
\begin{align*}\sum_{r,s,t} \frac{q^{r^2+rs+s^2+st+t^2}}{(q)_r (q)_s (q)_t} z_1^{r+s} z_2^{s+t} = ...

**6**

votes

**2**answers

669 views

### Closed form or/and asymptotics of a hypergeometric sum

Dear mathematicians,
I am a computer scientist wandering in the deep sea of combinatorics and asymptotics to pursue a recent interest in average case analysis of algorithms. In doing so, I designed ...

**5**

votes

**0**answers

259 views

### When does a triangle of numbers have a zero row sum?

Suppose we have a triangle of numbers defined by the recurrence relation
$$\left| n \atop k \right| = f(n,k) \left| n-1 \atop k \right| +g(n,k) \left| n-1 \atop k-1 \right| + [n=k=0],$$
for some ...

**1**

vote

**0**answers

291 views

### Transfinite Sums Related to a Sequence

Hello,
Given a sequence $S$ indexed by the finite ordinals, a limit ordinal $\alpha$, and $k \in \mathbb{N}$, define $S_{\alpha+k}$(the extension of $S$ to $\alpha+k$) to be the sum over the products ...

**8**

votes

**2**answers

1k views

### Expressions involving Eulerian numbers of the second kind: trying to show $\sum_{m=0}^{n} (-1)^m(m)m!(2n-m-2)!\left\langle\left\langle n\atop m\right\rangle\right\rangle\neq0$ for even $n$.

Considering the success of a previous question involving Eulerian numbers, I thought I might throw this question into the mix. It comes from some localization computations in GW theory, but in this ...

**17**

votes

**3**answers

2k views

### Need help proving that $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$

Hello.
I have been trying very hard to show that $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$ and could not quite get anywhere. This inequality has been verified by computer ...