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I have a simple set of polynomials which I know how to construct for each integer $n$, but I havn't been able to write them down in terms of concrete sums and products.

For $n\in\mathbb N_+$, we have the elements $\{\alpha_m\}_{m=1}^n$ and $\{\beta_m\}_{m=2}^n$, where everything in non-commutative. The polynomial $P_n(\alpha, \beta)$ is constructed by the following rules:

  1. We must have all combination of products of $\alpha$ and $\beta$'s, where we start with $\alpha_n$ or $\beta_n$ and multiply successively by $\alpha_m$ or $\beta_m$ on the right, where $m.
  2. To the right of any $\alpha_m$ we can only have $\alpha_{m-1}$ or $\beta_{m-1}$ (index goes down by 1).
  3. To the right of any $\beta_m$ we can only have $\alpha_{m-2}$ or $\beta_{m-2}$ (index goes down by 2).
  4. The product stops with $\alpha_1$ or $\beta_2$. All these terms are summed together.

For example

$ P_1(\alpha,\beta) = \alpha_1$ $ P_2(\alpha,\beta) = \alpha_2\alpha_1 + \beta_2$ $ P_3(\alpha,\beta) = \alpha_3\alpha_2\alpha_1 + \beta_3 \alpha_1 + \alpha_3\beta_2$ $ P_4(\alpha,\beta) = \alpha_4\alpha_3\alpha_2\alpha_1 + \alpha_4\alpha_3\beta_2 + \alpha_4\beta_3\alpha_1 + \beta_4\alpha_2\alpha_1 + \beta_4\beta_2$ $ P_5(\alpha,\beta) = 8-terms$

I haven't been able to write this polynomial in terms of concrete sums and product. I am only able to do it if point 3 is changed such that the index goes down with one. Do you have any ideas?


EDIT: Let me clarify what I want to achieve. Lets add $\beta_1$ and change point (3) such that the index only goes down by 1 at each product. In this case I think one can express the Polynomial as

$ Q_n(\alpha,\beta) = \sum_{\alpha=1}^n\prod_{i=1}^{\alpha}\left(\prod_{j=1}^i\sum_{M_{j-1}}\sum_{N_{j}}\right)\left(\prod_{m_i=N_i+1}^{M_i}\beta_{m_i}\prod_{n_i=M_{i-1}+1}^{N_i}\alpha_{n_i}\right)$

where $M_0 = 0$, $M_n = n$ and $ 0\leq N_i \leq n-2(\alpha-1),$ $ M_{i-1}+1\leq N_i \leq n-2(\alpha-i),\;\;\;\;\text{for}\;\; i=2, \dots, \alpha,$ $ N_i+1\leq M_i \leq n-2(\alpha-i)+1,\;\;\;\;\text{for}\;\; i=1, \dots, \alpha.$

I would like to have a formula like this for $P(\alpha,\beta)$, but I don't seem to be able to do that. It seems that the parity of $n$ plays an annoying role.

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    @Mark, If you follow the link in my comment above you will see that this is the exact recursion relation I am trying to solve. I will edit my question and make it more clear what I am trying to achieve.2012-03-26

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