Solving some problems of advanced analytical nature posed in the SIAM Review. II (Q2365222)

The author solves three problems: (1) Find closed expressions for \(S(v)\) and \(C(v)\), where \[ S(v):= \sum^\infty_{m=0} \sum^\infty_{n=1}(-1)^{m+n}(m^2+n^2)^{-1/2}\sin 2v(m^2+n^2)^{1/2} \] and \(C(v)\) is defined with cos in place of sin. (2) Find a closed expression \[ S_\nu(v):=\sum^\infty_{m=0} \sum^\infty_{n=1}(-1)^{m+n}(m^2+n^2)^{-\nu/2} J_\nu(2v(m^2+n^2)^{1/2}),\quad v\geq 0. \] (3) Determine the sum \[ \sum^n_{m=0}-{{1\over 4}\choose m}^2-{{1\over 4}\choose n-m}^2. \] Contrary to the previous problems now the solution is simple and quotable, it is \[ (-1)^n-{{1\over 2}\choose n}^3. \] For Part I see Simon Stevin 67, No. 3-4, 187-217 (1993; Zbl 0827.00005).