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

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

    True or false: \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 n and c_{\alpha} are some positive integers depending on n.
    Last edited by NonCommAlg; September 19th 2009 at 11:37 PM.
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