  Euler--Maclaurin and Poisson analogues of the summations $\sum_{a < n \leq b}
\chi(n) f(n)$, $\sum_{a < n \leq b} d(n) f(n)$, $\sum_{a < n \leq b} d(n) \chi
(n) f(n)$ have been obtained in a unified manner, where $(\chi (n))$ is a
periodic complex sequence; $d(n)$ is the divisor function and $f(x)$ is a
sufficiently smooth function on $[a,b]$. We also state a generalised Abel's
summation formula, generalised Euler's summation formula and Euler's summation
formula in several variables.
