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

• URL: http://arxiv.org/abs/2411.10126v3
• Date: Thu, 20 Mar 2025 10:14:47 GMT
• Title: Infinite series involving special functions obtained using simple one-dimensional quantum mechanical problems
• Authors: Sonja Gombar, Milica Rutonjski, Petar Mali, Slobodan Radošević, Milan Pantić, Milica Pavkov-Hrvojević,
• Abstract summary: In this paper certain classes of infinite sums involving special functions are evaluated analytically.<n>$L_nu2n+1-nuleft(fracnu+22;frac32;frac12right)$ is generalized hypergeometric function, $L_nu2n+1-nuleft(fracnu+22;frac32;frac12right)$ is calculated for integer $nu$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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