Local Frobenius Analysis of the Sequential Conformable Heun Equation at Three Finite α-Singular Points
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Conformable derivative, sequential conformable equation, Heun equation, Frobenius method, regular α-singular point, recurrence relation
Abstract
In this study, a specific Frobenius framework is proposed for the sequential conformable Heun equation in the original variable. The corresponding substitution z = xα, y(x) = Y (z) determines the relevant scaled parameters and establishes precise equivalence with the ordinary general Heun equation. The local analysis in x is carried out using the sequential operator Tα2 = Tα ◦ Tα with conformable steps xα, xα - 1, and xα - a. For 0 < α ≤ 1, a > 0, and a , 1, the finite singular points x = 0, x = 1, and x = a
1α
are proved to be regular α-singular. Each center yields characteristic exponents, the indicial equation, a three-term coefficient recurrence, nonresonant Frobenius fundamental conditions, and a guaranteed local convergence domain. To solve resonant cases, a recursive compatibility condition is used: if the condition is satisfied, a second ordinary conformable Frobenius series occurs; alternatively, the second independent solution contains a logarithmic term. Furthermore, sufficient and necessary termination conditions are obtained for finite conformable Frobenius solutions, as well as a determinant condition regarding the accessory parameter and an exact terminating example based on an identically zero residual. A numerical example shows how to calculate local coefficient sequences and evaluate the reduced series with a normalized conformable residual. When α = 1, the equation is reduced to the normal general Heun-Frobenius structure. The framework that is generated gives a unified original-variable description of the local solution framework at all three finite singular points.
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Copyright (c) 2026 Isam Najemadeen,and Erdal Bas
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