Analogues of Euler and Poisson Summation Formulae

[摘要] Euler–Maclaurin and Poisson analogues of the summations $sum_{a < n ≤ b}𝜒(n)f(n), sum_{a < n ≤ b}d(n) f(n), sum_{a < n ≤ b}d(n)𝜒(n) f(n)$ have been obtained in a unified manner, where (𝜒(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.

[学科分类] 数学（综合）

[关键词] Abel’s summation formula;Euler summation formula;Euler–Maclaurin summation formula;Poisson’s summation formula;Fourier series.