Analogues of Euler and Poisson summation formulae

Abstract

Euler--Maclaurin and Poisson analogues of the summations ${\sum }_{a<n\le b}\chi (n)f(n)$, ${\sum }_{a<n\le b}d(n)f(n)$, ${\sum }_{a<n\le b}d(n)\chi (n)f(n)$ 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.

9 pages