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Help me please to find a general coefficient $a_j$ of the following series $$ \left(\sum_{j=0}^{\infty}\frac{1}{j!}\left(\frac{t^2}{8p}\right)^j\right)\left(\sum_{j=0}^k\frac{(-1)^jt^{2j}}{4^jp^{2j}j!(n+j)!}\right)=1+\sum_{j}\frac{a_jt^{2j}}{p^j}. $$ Here $n \in N, p\geq2, k\in N$.

Thank you for your help.

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What is $k$? Should it be $\infty$? – Brian M. Scott Jun 29 '12 at 18:50
$k\in N$. I should take a partial sum. But we can think that $k$ is infinity. – Michael Jun 29 '12 at 20:15
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The coefficient of $t^{2m}$ in $\displaystyle\sum_{i=0}^\infty \dfrac{t^{2i}}{i!(8p)^i} \sum_{j=0}^\infty \dfrac{(-1)^j t^{2j}}{4^j p^{2j} j!(n+j)!}$ is, according to Maple, $\dfrac{\text{LaguerreL}(m,n,2/p)}{(8p)^m (m+n)!}$, where $\text{LaguerreL}(m,n,t)$ is a generalized Laguerre polynomial.

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Thank you. I am not familiar with the Laguerre polynomials and wikipedia give a bit different representation $(L_n^{(\alpha)}(x))$. I am confused what is $m, n and 2/p$ in it. Could you please give me a formula. Thank you very much. – Michael Jun 29 '12 at 21:25
LaguerreL($n,\alpha,x$) in Wikipedia's notation is $L_n^{(\alpha)}(x)$ – Robert Israel Jun 29 '12 at 23:39
Thank you very much. – Michael Jun 30 '12 at 14:06

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