Mathematical definition of Vigenère cipher

Assume a sequence of plaintext letters $P=p_0,p_1,p_2,\dots ,p_{n-1}$ and a key consisting of the sequence of letters $K=k_0,k1,k_2,\dots ,k_{m-1}$. Where $m$ typically is smaller than $n$.

A sequence of ciphertext letters:

$$\begin{array}{rl}C& =C_0,C_1,C_2,\dots ,C_{n-1}\\ & =E(K,P)\\ & =E((k_0,k_1,k_2,\dots ,k_{m-1}),(p_0,p_1,p_2,\dots ,p_{n-1}))\\ & =(p_0+k_0)\text{ mod }26,(p_1+k_1)\text{ mod }26,\dots \\ & \\ C_i& =(p_i+k_{i\text{ mod }m})\text{ mod }26\end{array}$$

For decryption, $p_i=(C_i-k_{i\text{ mod }m})\text{ mod }26$.