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

让 $\mathrm{一世}\subseteq \mathbb{电阻}$ 是一个区间和 $\text{β}：\mathrm{一世}\to \mathrm{一世}$ 具有唯一不动点的严格递增连续函数 ${秒}_0\in \mathrm{一世}$ 满足 $({秒}_0-吨)\left(\text{β}\right(吨)-吨)\ge 0$ 对所有人 $吨\in \mathrm{一世}$，其中等式仅在 $吨={秒}_0$. Hamza等人定义的一般量子算符。,$D_{\text{β}}\left[F\right]\left(吨\right):=\frac{F(\text{β}\left(吨\right))-F\left(吨\right)}{\text{β}\left(吨\right)-吨}$ 如果 $吨\ne {秒}_0$ 和 $D_{\text{β}}\left[F\right]\left({秒}_0\right):=F^{\mathrm{\prime }}\left({秒}_0\right)$ 如果 $吨={秒}_0,$推广 Jackson q -operator$D_q$ 还有哈恩（量子导数）算子， $D_{q,\omega }$. 关于与上述算子相关的β -Sturm-Liouville 特征值问题$D_{\text{β}}$，我们构造β-拉格朗日恒等式，证明它在${\mathcal{升}}_{\text{β}}^2\left(\right[\mathrm{一种},乙\left]\right),$ 并展示相应的特征值和特征函数的一些性质。