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Suppose $x_i$ are variables, $i\in \mathbb{Z}$, $n\in \mathbb{N}$.

I'm interested in solving the mixed recurrence relation:

$a_{i,n}=a_{i-1,n-1}+x_i \cdot a_{i,n-1}+a_{i+1,n-1}$ with initial conditions $a_{0,1}=x_i$, $a_{1,1}=a_{-1,1}=1$, and $a_{i,1}=0$ for every $i\neq -1,0,1$. (For each $i$, $a_{i,n}$ is a sequence of polynomials and the index of the sequence is $n$).

Is there any way to approach this problem?

joeyg
  • 511
  • The case $x=1$ is the Delannoy numbers https://en.wikipedia.org/wiki/Delannoy_number – Donald Splutterwit Aug 26 '17 at 20:21
  • It might be useful that the simplified system $$a_{i+1, n}=a_{i,n-1}+a_{i,n+1}$$ $$a_{1, 1}=a_{1,-1}=1$$ $$a_{1, i}=0\ \ \forall i\neq 1,-1$$ generates Pascal's Triangle. – Kajelad Aug 27 '17 at 05:27

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