I read the paper On some strange summation formulas by R. William Gosper which includes $$\sum_{k = 1}^{\infty}\frac{(-1)^k\cos\big(\sqrt{k^{2}\pi^{2} -a^2})}{k^{2}}=\frac{\pi^2}{12}\big({-\cosh(a)}+\frac{3}{a}\sinh(a)\big)$$ I was looking at the following series maple could sum. Any idea how to prove it, thanks. Define $$S_1(a)=\sum _{k=1}^{\infty } \frac{(-1)^k \cos \left(\sqrt{k^2\pi ^2-a}\right)}{\left(k^2+1\right)}$$ $$S_2(a)=\sum _{k=1}^{\infty } \frac{(-1)^k \cos \left(\sqrt{k^2\pi ^2-a}\right)}{\left(k^2+1\right)^2}$$ Let $\beta=\sqrt{-a-\pi^2}.\,$ Are the following true?:
$$S_1(a)=\frac{1}{2} \big(\pi\, \text{csch}(\pi ) \cos ( \beta)-\cosh \left(\sqrt{a}\right)\big)$$ $$S_2(a)=\frac{1}{4} \Big(\pi\, \text{csch}(\pi ) \cos ( \beta) \big[1+\pi \coth (\pi )\big] -2 \cosh \left(\sqrt{a}\right)-\frac{\pi ^3 \text{csch}(\pi ) \sin ( \beta)}{ \beta}\Big)$$