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m Specifying the dimension of x |
specifying the index of summation |
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&= \exp(\mathbf{x}^\top \mathbf{x'}) \exp( - \frac{1}{2}\|\mathbf{x}\|^2) \exp( - \frac{1}{2}\|\mathbf{x'}\|^2) \\
&= \sum_{j=0}^\infty \frac{(\mathbf{x}^\top \mathbf{x'})^j}{j!} \exp\left(-\frac{1}{2}\|\mathbf{x}\|^2\right) \exp\left(-\frac{1}{2}\|\mathbf{x'}\|^2\right)\\
&= \sum_{j=0}^\infty \sum_{\sum\limits_{i}^k n_i=j}
\exp\left(-\frac{1}{2}\|\mathbf{x}\|^2\right)
\frac{x_1^{n_1}\cdots x_k^{n_k} }{\sqrt{n_1! \cdots n_k! }}
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