The heat kernel on a smooth manifold $M$ is defined to be the solution of $$(\Delta_x-\partial_t)H(x,y,t)=0,\qquad \text{in}~ M\times M\times (0,+\infty)$$ with initial data $\lim_{t\to 0}H(x,y,t)=\delta_x(y)$ in the sense of distribution, where $\delta_x(y)$ is the point mass at $x.$
I have already known that the heat kernel on $\mathbb{S}^1$ is $$H(x,y,t)=\frac{1}{2\pi}+\frac{1}{\pi}\sum_{k=1}^\infty e^{-k^2t}\cos k(x-y)\tag{1}$$ occasionally I find in Heat kernels... that the heat kernel on $\mathbb{S}^1$ can also be written as $$H(x,y,t)=\frac{1}{\sqrt{4\pi t}}\sum_{k\in\mathbb{Z}}e^{-\frac{(x-y-2k\pi)^2}{4t}}\tag{2}$$ so how can I show that they are in fact the same thing? I wonder this is related to the uniqueness of solution, but I am not good at PDEs, can someone help me?