Fuchs–Frobenius

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1: 31.18 Methods of Computation

… ►Independent solutions of (31.2.1) can be computed in the neighborhoods of singularities from their Fuchs–Frobenius expansions (§31.3), and elsewhere by numerical integration of (31.2.1). …

2: 31.3 Basic Solutions

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§31.3(i) Fuchs–Frobenius Solutions at $z=0$

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§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities

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3: 31.11 Expansions in Series of Hypergeometric Functions

… ►Let $w(z)$ be any Fuchs–Frobenius solution of Heun’s equation. …The Fuchs-Frobenius solutions at $\mathrm{\infty }$ are … ►Every Fuchs–Frobenius solution of Heun’s equation (31.2.1) can be represented by a series of Type I. …Then the Fuchs–Frobenius solution at $\mathrm{\infty }$ belonging to the exponent $\alpha$ has the expansion (31.11.1) with … ►Such series diverge for Fuchs–Frobenius solutions. …

4: 2.7 Differential Equations

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§2.7(i) Regular Singularities: Fuchs–Frobenius Theory

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5: 31.10 Integral Equations and Representations

… ►Fuchs–Frobenius solutions $W_m(z)={\tilde{\kappa }}_m{z}^{-\alpha }H\mathrm{\ell }(1/a,q_m;\alpha ,\alpha -\gamma +1,\alpha -\beta +1,\delta ;1/z)$ are represented in terms of Heun functions $w_m(z)=(0,1){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$ by (31.10.1) with $W(z)=W_m(z)$, $w(z)=w_m(z)$, and with kernel chosen from …