Fuchs%E2%80%93Frobenius theory
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1: 31.18 Methods of Computation
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►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).
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2: 16.23 Mathematical Applications
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►A variety of problems in classical mechanics and mathematical physics lead to Picard–Fuchs equations.
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§16.23(iv) Combinatorics and Number Theory
…3: 31.3 Basic Solutions
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§31.3(i) Fuchs–Frobenius Solutions at
… ►§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities
…4: 31.11 Expansions in Series of Hypergeometric Functions
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►Let be any Fuchs–Frobenius solution of Heun’s equation.
…The Fuchs-Frobenius solutions at are
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►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 belonging to the exponent has the expansion (31.11.1) with
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►Such series diverge for Fuchs–Frobenius solutions.
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5: 2.7 Differential Equations
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§2.7(i) Regular Singularities: Fuchs–Frobenius Theory
… ►In theory either pair may be used to construct any other solution …6: Bibliography F
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Theory and Computation of Spheroidal Harmonics with General Arguments.
Master’s Thesis, The University of Western Australia, Department of Physics.
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Application of the -function theory of Painlevé equations to random matrices: PIV, PII and the GUE.
Comm. Math. Phys. 219 (2), pp. 357–398.
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Application of the -function theory of Painlevé equations to random matrices: , , the LUE, JUE, and CUE.
Comm. Pure Appl. Math. 55 (6), pp. 679–727.
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Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen.
Math. Ann. 63 (3), pp. 301–321.
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Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond.
In Recent Perspectives in Random Matrix Theory and Number Theory,
London Math. Soc. Lecture Note Ser., Vol. 322, pp. 31–78.
7: 31.10 Integral Equations and Representations
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►Fuchs–Frobenius solutions are represented in terms of Heun functions by (31.10.1) with , , and with kernel chosen from
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8: 8.23 Statistical Applications
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►In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of ; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319).
9: 27.2 Functions
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§27.2(i) Definitions
… ►This is the number of positive integers that are relatively prime to ; is Euler’s totient. … ►
27.2.8
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27.2.9
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10: Bibliography H
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The Laplace transform for expressions that contain a probability function.
Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).
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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions.
Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).
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Integrals that contain a probability function of complicated arguments.
Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).
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Sums with cylindrical functions that reduce to the probability function and to related functions.
Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).
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Some properties and applications of the repeated integrals of the error function.
Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.
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