Does there exist a closed form for the non-integer shifted sinc-function series: $\frac{\sin(n+a)x}{(n+a)x}$?

Question

Here I want to get the closed form solution of the following summation

$$ \sum_{n=1}^\infty \frac{\sin(n+a)}{n+a}, a\in \mathbb{R}^+ \qquad(1) $$

Or the more general form:

$$ f(a;x) =\sum_{n=1}^\infty\operatorname{sinc} [(n+a)x]= \sum_{n=1}^\infty \frac{\sin [(n+a)x]}{(n+a)x}, a,x\in \mathbb{R}^+ \qquad(2) $$

I looked up into several books and got a similar seires with its closed form:

1. Equation (550) in Summation of Series 2nd ed. (written by Jolley). $$ f(1/2;x)= \sum_{n=1}^\infty \frac{\sin [(n+1/2)x]}{(n+1/2)x} = \frac1x\left(\frac{\pi}{2}-2\sin\frac x2\right),0<x<\pi \qquad $$

But 2 issues in this series: (A) the domain of $x$ is limited (B) the parameter $a$ is limited.

2. Equation (551) in the same book

$$ \sum_{n=-\infty}^\infty \frac{\sin [(n-a)x]}{(n-a)x} = \frac{\pi}{x},0<x<2\pi \qquad $$ Also the domain of $x$ is limited. And the summation starts from negative infinity.

Anyone can help me?

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Possible relating QUESTIONS:

Does there exist a closed form for the sinc function series $\sum_{n=1}^\infty \frac{\sin\sqrt{n^2+1}}{\sqrt{n^2+1}}$?

Answer 1

$$I=\sum_{n=1}^\infty \frac{e^{+i (n+a)x}}{ (n+a)x}=\frac{e^{+i (a+1) x} }{(a+1) x}\, _2F_1\left(1,a+1;a+2;e^{+i x}\right)$$ $$J=\sum_{n=1}^\infty \frac{e^{-i (n+a)x}}{ (n+a)x}=\frac{e^{-i (a+1) x} }{(a+1) x}\, _2F_1\left(1,a+1;a+2;e^{-i x}\right)$$ $$\sum_{n=1}^\infty\frac{\sin((n+a)x)}{(n+a)x}=\frac{I-J}{2i}$$

Answer 2

$$\begin{eqnarray*}\sum_{n\geq 1}\frac{\sin(a+n)}{a+n}&=&\text{Im}\int_{0}^{+\infty}\sum_{n\geq 1}\exp\left(ia+in-ax-nx\right)\,dx\\ &=& \int_{0}^{+\infty}\frac{\sin(a)e^{-x}-\sin(a+1)}{2\left(\cos(1)-\cosh(x)\right)}\,e^{-ax}\,dx\\&=&\int_{0}^{1}\frac{u^a\left(\sin(a+1)-u\sin(a)\right)}{1-2u\cos(1)+u^2}\,du\end{eqnarray*}$$ can be computed in a explicit way for any $a\in\mathbb{Q}^+$ by partial fraction decomposition.
The asymptotic behaviour of the LHS is simple to study through the integral representation, but a closed form for any $a\in\mathbb{R}^+$ is beyond reach, if not through hypergeometric ${}_2 F_1$ functions. This is reminiscent (and not by chance) of the Hurwitz zeta function and the Digamma function.