Infinite series involving special functions obtained using simple one-dimensional quantum mechanical problems (2411.10126v3)

Abstract: 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 $\sum^{\infty}_{n=0}\frac{2^{2n}}{(2n+1)!}\Gamma^{2}\left(n+\frac{3}{2}\right)\left[\hspace{0.2mm}_2\hspace{-0.03cm}F_1\left(-n,\frac{\nu+2}{2};\frac{3}{2};\frac{1}{2}\right)\right]^{2}$, $\sum^{\infty}{n=0}\frac{\left[L{\nu}^{2n+1-\nu}\left(\frac{b^{2}}{2}\right)\right]^{2}b^{4n}}{2^{2n}(2n+1)!}$ and $\sum^{\infty}{n=1}\frac{\big[J{\nu+1}(n\pi)\big]^{2}}{n^{2\nu}}$, where $2\hspace{-0.03cm}F_1\left(-n,\frac{\nu+2}{2};\frac{3}{2};\frac{1}{2}\right)$ is generalized hypergeometric function, $L{\nu}^{2n+1-\nu}\left(\frac{b^{2}}{2}\right)$ associated Laguerre polynomial and $J_{\nu+1}(n\pi)$ Bessel function of the first kind, are calculated for integer $\nu$. 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 $\sum_{n=1}^{\infty}\frac{\left[\mathsf{H}_{\nu}(n\pi)\right]^{2}}{n^{2\nu}}$ containing Struve functions of the first kind $\mathsf{H}_{\nu}(n\pi)$ 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.