# Compute the sum given a recursion for each summand

I have a system of two recursive equations with two unknowns A[n] and B[n] which apparently Mathematica can't solve

RSolve not evaluating when given my system of recurrence equations

Each $$A[n]$$ is a polynomial divided by a Pochhammer symbol: $$A[n] = p(q)/(q)_n = p(q)/\prod_{j=1}^n (1-q^j)$$ where $$p(q) \in \mathbb{Z}[q]$$.

I am only interested in computing the formal sum $$\sum_{n \geq 0} A[n]$$ as a power series in $$q$$. Is this something Mathematica can do? I know how to get the corresponding polynomial to each order of $$n$$ but I am interested in a closed formula for the formal sum.

For completeness I am copying the system of equations with initial conditions here:

RSolve[
{(1 - q^n) A[n] ==  q^(3 n - 1) B[n - 1] + q^(2 n + 1) A[n - 1] +  q^(4 n - 2) A[n - 2] + q^(4 n - 2) B[n - 2],
B[n] ==  q^(n + 1) A[n - 1] - q^(5 n - 5) A[n - 3] - q^(5 n - 5) B[n - 3],
A[0] == 1, A[1] == q^3/(1 - q), A[2] == q^6/((1 - q) (1 - q^2)),
B[0] == 0, B[1] == q^2, B[2] == q^6/(1 - q)},
{A[n], B[n]}, n]

• What is _ 2 supposed to mean? Typo? – MikeY Feb 26 at 17:09
• is the Pochhammer symbol defined in the fifth line, but I'll expand it anyway thanks – Reimundo Heluani Feb 26 at 17:12
• no, the coefficients are actual powers, I'll copy and paste from my other question the system. – Reimundo Heluani Feb 27 at 9:57
• It didn't work for me. You might consider an add-on package from the folks at RISC, www3.risc.jku.at/research/combinat/software/ergosum/RISC/… – MikeY Feb 27 at 13:44