Deriving Feature Map formula for Inhomogeneous Polynomial Kernel
Let $x, y \in \mathbb R^N, \mathbf \alpha \equiv (\alpha_0, \dots, \alpha_N) \equiv (\alpha_0, \mathbf \alpha'_0), \; \mathbf z^\alpha \equiv z_0^{\alpha_0} \dots z_N^{\alpha_N}$ and ${d \choose \alpha}$ be the multinomial coefficient. Then

$$\left(\left&lt;\mathbf x, \mathbf y \right&gt; + c\right)^d = \sum_{|\mathbf \alpha| = d} {d \choose \alpha} c^{\alpha_0} (\mathbf x \mathbf y)^{\mathbf \alpha'_0}$$ 

If you redistribute the coefficient ${d \choose \alpha} c^{\alpha_0}$ between $\Phi(\mathbf x)$ and $\Phi(\mathbf y)$, it follows that

$$\Phi(\mathbf z) = \left(\sqrt{ {d \choose \alpha} c^{\alpha_0}} \mathbf z^{\alpha'_0} \right)_{\forall |\mathbf \alpha| = d}$$

References


A Tutorial on Support Vector Machines for Pattern Recognition (cf. 6.1. The VC Dimension for Polynomial Kernels)
Data Mining and Analysis: Fundamental Concepts and Algorithms, Ch.5, pp 149-151.