A $\beta$-Sturm Liouville problem associated with the general quantum operator

Abstract

Let $I\subseteq \mathbb{R}$ be an interval and $\beta :I\to I$ a strictly increasing and continuous function with a unique fixed point $s_0\in I$ which satisfies $(s_0-t)(\beta (t)-t)\ge 0$ for all $t\in I$, where the equality holds only when $t=s_0$. The general quantum operator defined in 2015 by Hamza et al., $D_{\beta }[f](t):=\frac{f(\beta (t))-f(t)}{\beta (t)-t}$ if $t\ne s_0$ and $D_{\beta }[f](s_0):=f^{\prime }(s_0)$ if $t=s_0,$ generalizes the Jackson $q$-operator $D_q$ and also the Hahn $(q,\omega )$-operator, $D_{q,\omega }$. Regarding a $\beta -$Sturm Liouville eigenvalue problem associated with the above operator $D_{\beta }$, we construct the $\beta -$Lagrange's identity, show that it is self-adjoint in ${\mathcal{L}}_{\beta }^2([a,b]),$ and exhibit some properties for the corresponding eigenvalues and eigenfunctions.

15 pages

References in corpus (1)

• On Basic Fourier-Bessel Expansions