Infinite series involving special functions obtained using simple one-dimensional quantum mechanical problems

In this paper certain classes of infinite sums involving special functions are evaluated analytically by application of basic quantum mechanical principles to simple models of half harmonic oscillator and a particle trapped inside an infinite potential well. The infinite sums sum^infty_n=0frac{2²ⁿ}{(2n+1)!}Γ²left\(n+frac{3}{2}right\)left\[hspace{0.2mm}_2hspace{-0.03cm}F_1left\(-n,frac{ν+2}{2};frac{3}{2};frac{1}{2}right\)right\]², sum^infty_n=0frac{left\[L_ν^2n+1-νleft\(frac{b²}{2}right\)right\]²b⁴ⁿ}{2²ⁿ(2n+1)!} and sum^infty_n=1frac{big\[J_ν+1(nπ)big\]²}{n^2ν}, where _2hspace{-0.03cm}F_1left\(-n,frac{ν+2}{2};frac{3}{2};frac{1}{2}right\) is generalized hypergeometric function, L_ν^2n+1-νleft\(frac{b²}{2}right\) associated Laguerre polynomial and J_ν+1(nπ) Bessel function of the first kind, are calculated for integer ν. It is also demonstrated that the same procedure can be generalized by application to some classes of functions which are not regular wave functions leading to additional infinite sums, as a consequence of which the series sum_n=1^inftyfrac{left\[mathsf{H}_ν(nπ)right\]²}{n^2ν} containing Struve functions of the first kind mathsf{H}_ν(nπ) are evaluated. Convergence of the evaluated series, additionally verified by the application of different convergence tests, is secured by the properties of the corresponding Hilbert space.