Analogues of Euler and Poisson summation formulae

Euler–Maclaurin and Poisson analogues of the summations $\sum_{a < n \leq b} \chi(n) f(n)$,$\sum_{a < n \leq b} d(n) f(n)$,$\sum_{a < n \leq b} d(n) \chi (n) f(n)$havebeenobtainedinaunifiedmanner,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.