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

TL;DR

The paper addresses the analytic evaluation of nontrivial infinite sums of special functions that arise in simple one-dimensional quantum mechanical problems. By applying expansion coefficients from eigenstates of a half harmonic oscillator and an infinite potential well, it derives closed-form identities for series involving $$_2F_1(-n,\cdot;\cdot;\cdot)$$, associated Laguerre polynomials, Bessel and Struve functions, and extends the method to non-regular wave functions to obtain additional sums. Key contributions include explicit ν-dependent formulas and special-case results such as $\frac{2\pi}{3\sqrt{3}}$ (for $\nu=1$ in the half oscillator) and $\frac{4}{15}$ (for $\nu=2$ in the infinite well), along with convergence verifications through ratio and comparison tests grounded in Hilbert space structure. The findings offer analytic tools linking quantum mechanical models to exact summation identities for a broad class of special functions, with potential applications in spectral sum rules and mathematical physics.

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 $\sum^{\infty}_{n=0}\frac{2^{2n}}{(2n+1)!}Γ^{2}\left(n+\frac{3}{2}\right)\left[\hspace{0.2mm}_2\hspace{-0.03cm}F_1\left(-n,\frac{ν+2}{2};\frac{3}{2};\frac{1}{2}\right)\right]^{2}$, $\sum^{\infty}_{n=0}\frac{\left[L_ν^{2n+1-ν}\left(\frac{b^{2}}{2}\right)\right]^{2}b^{4n}}{2^{2n}(2n+1)!}$ 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.

Figures (1)

• Figure 1: Wave functions \ref{['wf1']} and \ref{['wf3']} are shown in a) and c), respectively. Functions with discontinuity point $x=0$, \ref{['funkcgr']} and \ref{['wf4']}, are shown in b) and d). Functions \ref{['wf3']} and \ref{['wf4']} are plotted for $a=1$. Note that the functions shown in b) and d) do not represent regular wave functions since they are not continuous, but do, on the other hand, allow evaluation of additional infinite sums.