# Series identity

by mhill
Tags: identity, series
 P: 193 for every sequence of numbers a_n E_n is this identity correct ? $$\sum_{n= -\infty}^{\infty}a_n e^{2\pi i E_{n}}= \sum_{n= -\infty}^{\infty}a_n \delta (x-E_{n})$$
 P: 2,251 This is true $$\sum_{n=-\infty}^{+\infty} e^{i 2 \pi n x} = \sum_{k=-\infty}^{+\infty} \delta(x-k)$$ but to generalize it with arbitrary coefficients (that are placed on both sides) is not a true equality.
 P: 193 rbj and matt were right only this $$\sum_{n=-\infty}^{+\infty} e^{i 2 \pi n x} = \sum_{k=-\infty}^{+\infty} \delta(x-k)$$ (1) is correct , however my question is if using Fourier analysis we could generalized to an identity $$\sum_{n=-\infty}^{+\infty}a_{n} e^{i 2 \pi n x} = \sum_{k=-\infty}^{+\infty} b_{n}\delta(x-k)$$ where the a_n and b_n are related by some way , this is interesting regarding an article of Functional equation for Dirichlet series, using (1) the author was able to proof the functional equation for Riemann Zeta, my idea was to develop a functional equation for almost every dirichlet series to see where they have the 'poles'
 P: 193 If we have in the general case $$\sum_{n=-\infty}^{+\infty}b_{n} e^{i 2 \pi n x} = D A(x)$$ Where A(x) is the partial sum of a_n and D is the derivative operator , in case A(x)=[x] we recover usual delta identity , then i believe we can calculate b_n by the Fourier integral $$b_n = \int_{0}^{1} dx DA(x) e^{-2i\pi x}$$