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Does anyone know how to calculate the following summation, in which $a$, $b$ and $k$ are constant real numbers:

$$\sum_{n=-\infty}^{\infty} \frac{k n +a}{\big((k n +a)^2 + b^2\big)^{\frac{3}{2}}}$$

In the above relation, $n$ is an integer number which changes from $-\infty$ to $\infty$.

jvdhooft
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sara nj
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  • What are $a$ and $l$ and $b$? Are they integers too? – Franklin Pezzuti Dyer Jul 05 '17 at 15:47
  • yes, they are constant real numbers. @Nilknarf – sara nj Jul 05 '17 at 15:49
  • Do you know how can one calculate it? @Nilknarf – sara nj Jul 05 '17 at 15:50
  • No, I have no idea. Maybe try partial fractions? – Franklin Pezzuti Dyer Jul 05 '17 at 15:54
  • Why did you delete your previous almost same question? – Jaideep Khare Jul 05 '17 at 16:03
  • Because I decide to ask it in a simpler way. Could you hep? @JaideepKhare – sara nj Jul 05 '17 at 16:03
  • After playing around with this a bit, I'm now sure that the result of this sum cannot be written in terms of elementary functions. If you're looking for a nice concise closed form result, there is no such thing. The answer would be related to Hurwitz zeta functions $\zeta(3/2, a+ib)$ and $\zeta(1/2, a+ib)$. – Hamed Jul 05 '17 at 17:01
  • Could you please tell me what is the answer in terms of Hurwitz zeta functions? Any answer concise closed or complicated can help me@Hamed – sara nj Jul 06 '17 at 04:35
  • in Hurwitz zeta functions power of n is equal to zero but in my case, power is equal to 2. @Hamed – sara nj Jul 06 '17 at 04:41
  • Where is $k$ in Hurwitz zeta functions you wrote? – sara nj Jul 06 '17 at 04:45
  • Also, the function is not defined for s=1/2 – sara nj Jul 06 '17 at 05:11
  • I said a "variation" of Hurwitz. First off, you can factor $k$ out of the sum, so $k$ doesn't really matter all that much (rescale $a\to a/k$ and $b\to b/k$). Hurwitz zeta is $\zeta(s,q)=\sum_{n=0}^\infty 1/(n+q)^s$. What you need is two slight variations of it $\zeta_1(s,q)=\sum_{n=0}^\infty 1/|n+q|^s$ and $\zeta_2(s,q)=\sum_{n=0}^\infty (n+q)/|n+q|^{s+1}$. Point is, although niether of these sums are Hurwitz zeta itself, you cannot write them via elementary functions either. Note that these sums all are absolutely convergent to the same value $\zeta(s,q)$ absolutely converges to. – Hamed Jul 06 '17 at 05:22
  • Do you know any good approximation which may help to find the value of the summation? @Hamed – sara nj Jul 06 '17 at 06:47
  • If you're looking for a good approximation, you should consider posting another question. I'm sure there are people infinitely better than me when it comes to approximating sums. – Hamed Jul 06 '17 at 16:17

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