Define $ \Lambda_{i}(u) = \sum_{r=0}^{\infty} \Lambda_{i,r}u^{r}, \Psi_{i}(u) = \sum_{m=0}^{\infty} \psi_{i,m}u^{m}=k_i\frac{\Lambda_{i}(uq_i^{-1})}{\Lambda_{i}(uq_i)}. $ How can we show that $ \Lambda_{i,r}=\frac{1}{k_i(q_i^r-q_i^{-r})}\sum_{t=1}^{r}q_i^{r-t}\psi_{i,t}\Lambda_{i,r-t} $ and $ \psi_{i,r}=-k_i(q_i^r-q_i^{-r})\Lambda_{i,r}-\sum_{t=1}^{r-1}q_i^{r-t}\psi_{i,t}\Lambda_{i,r-t} ?$ Many thanks.
Compare of coefficients of two formal power series
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0It seems the index $i$ doesn't figure in the question? It would make the formulas a lot easier to read if you omit it. – 2011-07-20
1 Answers
I'll supress the index $i$ as suggested in my comment.
Multiplying through by $\Lambda(uq)$ yields
$\Psi(u)\Lambda(uq)=k\Lambda(uq^{-1})\;,$
or, expressed in terms of the series,
$\left(\sum_{m=0}^\infty \psi_mu^m\right)\left(\sum_{r=0}^\infty\Lambda_r(uq)^r\right)=k\sum_{r=0}^\infty\Lambda_r(uq^{-1})^r\;.$
The multiplication on the left leads to a convolution of the coefficient sequences, and comparing coefficients of the individual powers of $u$ on both sides gives
$\sum_{t=0}^rq^{r-t}\psi_t\Lambda_{r-t}=k\Lambda_rq^{-r}\;.$
Noting that $\psi_0=\Psi(0)=k\Lambda(0)/\Lambda(0)=k$, we can separate out the $t=0$ term and take it to the other side:
$\sum_{t=1}^rq^{r-t}\psi_t\Lambda_{r-t}=k\Lambda_r\left(q^{-r}-q^r\right)\;.$
Now solving for $\Lambda_r$ yields the first of your equations, up to a sign that one of us must have gotten wrong. For the second equation, additionally separate out the $t=r$ term and solve for $\psi_r$; however, you need to divide through by $\Lambda_0$ in the process, which isn't in your equation; are you assuming $\Lambda_0=1$?
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0thank you very much. Yes, $\Lambda_0=1$. – 2011-07-20