ABSTRACT In this paper, we have developed the spectral theory for a conformable fractional Sturm-Liouville problem with boundary conditions which include conformable fractional derivatives of order α, , and prove a completeness theorem and an expansion theorem. We obtain the canonical infinite product constructed by the zero set of the characteristic function of . Also, we calculate the regularized trace formula of the eigenvalues and investigate the inverse nodal problem for this problem with real-valued coefficients on a finite interval. The oscillation of the eigenfunctions for sufficiently large n is established, and an asymptotic formula for elements constructed by of nodal points is obtained. The uniqueness theorem is proved, and an effective procedure for solving the inverse problem is given. Finally, we present two examples to illustrate our theoretical findings.

中文翻译：

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适形分数Sturm-Liouville问题的迹公式和逆节点问题

摘要 在本文中，我们开发了具有边界条件的可整合分数 Sturm-Liouville 问题的谱理论，其中包括 α, 阶可整合分数阶导数，并证明了完备性定理和展开定理。我们得到由 的特征函数的零集构造的典型无穷积。此外，我们计算了特征值的正则化迹公式，并在有限区间上用实值系数研究了该问题的逆节点问题。建立了足够大的n的特征函数的振荡，并得到了由节点构成的单元的渐近公式。证明了唯一性定理，给出了求解逆问题的有效方法。最后，