arXiv2025
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 $\sum^{\infty}_{n=0}\frac{2^{2n}}{(2n+1)!}Γ^{2}\left(n+\frac{3}{2}\right)\left[\hspace{0.2mm}_2\hspace{-0.03cm}F_1\left(-n,\frac{ν+2}{2};\frac{3}{2};\frac{1}{2}\right)\right]^{2}$, $\sum^{\infty}_{n=0}\frac{\left[L_ν^{2n+1-ν}\left(\frac{b^{2}}{2}\right)\right]^{2}b^{4n}}{2^{2n}(2n+1)!}$ and $\sum^{\infty}_{n=1}\frac{\big[J_{ν+1}(nπ)\big]^{2}}{n^{2ν}}$, where $_2\hspace{-0.03cm}F_1\left(-n,\frac{ν+2}{2};\frac{3}{2};\frac{1}{2}\right)$ is generalized hypergeometric function, $L_ν^{2n+1-ν}\left(\frac{b^{2}}{2}\right)$ 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 $\sum_{n=1}^{\infty}\frac{\left[\mathsf{H}_ν(nπ)\right]^{2}}{n^{2ν}}$ 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.