While messing around with summations of the reciprocals of quadratics from $n=0$ to $\infty$ on wolfie.
I made this discovery: $$ \sum_{n=0}^{\infty}\frac{1}{(an)^2+b^2}=\frac{1}{2ab^2}(a+b\coth{\frac{b\pi}{a}})$$ I am not sure why this is true... I also want to know if there is a closed form for a general quadratic: $$\sum_{n=0}^{\infty}\frac{1}{an^2+bn+c}$$ Can anyone shed some light on the connection between summations and hyperbolic trigonometric functions? Are there any theoretical explanations or mathematical theories that help us understand why such a connection exists? I'm eager to learn more and would greatly appreciate any insights or references that can help me delve deeper into this mathematical phenomenon.
I want to use this knowledge to figure out this summation as it is the result of an integral I'm solving: $$ \sum_{n=0}^{\infty}\frac{1}{2n^2+2n+1}$$