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Thread: n-th derivative of e^y and the partition function

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    n-th derivative of e^y and the partition function

    Suppose $\displaystyle y=f(x)$ is $\displaystyle n$-times differentiable on some interval. For any partition $\displaystyle \alpha: a_1 + \cdots + a_k = n, \ a_j \geq 1,$ define $\displaystyle y^{\alpha}=\prod_{j=1}^k \frac{d^{a_j}y}{dx^{a_j}}.$

    Example: for $\displaystyle n=5$ and the partitions $\displaystyle \alpha: 1 + 1 + 1 + 2 = 5$ and $\displaystyle \beta: 2+3=5$ we have $\displaystyle y^{\alpha}= (y')^3y''$ and $\displaystyle y^{\beta}=y''y'''.$

    True or false: $\displaystyle \frac{d^n e^y}{dx^n}=\left(\sum_{\alpha} c_{\alpha}y^{\alpha} \right)e^y,$ where the sum is over all the partitions of $\displaystyle n$ and $\displaystyle c_{\alpha}$ are some positive integers depending on $\displaystyle n.$
    Last edited by NonCommAlg; Sep 19th 2009 at 11:37 PM.
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