Frobenius-Polynomial Second-Order Linear Differential Equations: Structure and Recognition in the Real-Indicial Case

We study equations (Formula presented.), with (Formula presented.), that admit two linearly independent local solutions (Formula presented.) and (Formula presented.) at the regular singular point 0, where A and B are polynomials and (Formula presented.). We call these Frobenius-polynomial type ( (Formula presented.) ) equations and focus on the subclass for which both roots of the indicial equation at (Formula presented.) are real. Writing (Formula presented.) and (Formula presented.), we derive (Formula presented.) and (Formula presented.). Factoring R determines every pole and residue of P, while its vanishing order at 0 distinguishes the possible exponent representations. We prove that the degree equation at ∞ has two distinct real roots. For each solution exponent, subtracting that exponent from the two roots leaves at most two non-negative-integer candidate degrees for its polynomial factor. For proper rational functions P and Q whose coefficients are algebraic over (Formula presented.), we give a recognition algorithm and prove its termination, soundness, and completeness for the real-indicial subclass, including the resonant case. The final step tests the existence and, in the common-exponent case, the linear independence of the required polynomial solutions by solving finitely many homogeneous linear systems for their coefficients. Three Calogero-type families, a rational example outside those families, and a negative general-Heun-type example illustrate the procedure.