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5
votes
0answers
271 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 ...
4
votes
0answers
291 views

Generalization of Tamarkin’s ARO 1993, final round, problem 10/8: part II

Let us use the notations of my previous question about Tamarkin's problem. Let $\ell\in\left\lbrace 0,1,...,p\right\rbrace$. An element $f\in \mathbb Z^{\mathbb Z}$ is said to be ...
2
votes
0answers
95 views

system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that $$a_1^3+a_2^3+a_3^3+a_4^3=0$$ $$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$ ...
2
votes
0answers
71 views

Multiplying three factorials with three binomials in polynomial identity

I have checked the following identity (1) below for $n\leq 40$ with a computer. Let $(n)_k$ denote the falling factorial $n(n-1)\ldots (n-k+1)$, let $Z_n=\sum_{k=0}^n (n)_k x^{n-k}$, and finally let ...
1
vote
0answers
311 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 ...