I Have seen two versions of the Polynomial Kernel during my time learning Kernel Methods for things such as regression analysis.

1) $\kappa_d(x,y) = (x \cdot y)^d$

2) $\kappa_d(x,y) = (x \cdot y + 1)^d$source

Without knowing deeply the mathematics behind these things, I attempted a proof of a polynomial kernel function that produces the kernel with all other lower-order polynomial terms (I set $x_i \rightarrow (x_i,1)$) and came out with 2).

Is this correct?

What Mathematics must I know to perform a rigorous proof of there being such a Kernel?