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

We study equations y′′+P(x)y′+Q(x)y=0, with P,Q∈C(x), that admit two linearly independent local solutions y1=xrA(x) and y2=xsB(x) at the regular singular point 0, where A and B are polynomials and A(0)B(0)≠0. We call these Frobenius-polynomial type (FP) equations and focus on the subclass for which both roots of the indicial equation at x=0 are real. Writing W(y1,y2)=xr+s−1R(x) and W(y1′,y2′)=xr+s−3R1(x), we derive P(x)=−(r+s−1)/x−R′(x)/R(x) and Q(x)=R1(x)/(x2R(x)). 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 Q, 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.