2
votes
1answer
264 views

Difference equation $A(n,x)=p(x)A(n-1,x-1)+q(x)A(n-1,x)$

I asked this question on MSE, but didn't get enough information. If it is a violation of some norms, let me know, I'll delete it. I'm having problem solving this difference equation. Initially I ...
5
votes
0answers
261 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 ...
9
votes
3answers
1k views

Solving a general two-term combinatorial recurrence relation

What is known about explicit (not necessarily closed-form) solutions to the recurrence $$R^n_k= (\alpha n) R^{n-1}_k + (\alpha' n + \beta' k) R^{n-1} _{k-1},$$ with initial condition $R_0^0 = 1$ and ...
2
votes
1answer
2k views

Solving partial difference equation

I am trying to solve the following partial difference equation: $$A_k^{n+1}=(k+1)A_{k+1}^n+(n+2-k)A_{k-1}^n $$ with initial condition: $$\begin{cases} A_0^0&=1\\ A_1^0&=1 \end{cases}$$ I ...