classification of cases
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1: 2.8 Differential Equations with a Parameter
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§2.8(i) Classification of Cases
…2: 18.27 -Hahn Class
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►A (nonexhaustive) classification of such systems of OP’s was made by Hahn (1949).
…The generic (top level) cases are the -Hahn polynomials and the big -Jacobi polynomials, each of which depends on three further parameters.
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►In case of unbounded sequences (18.27.2) can be rewritten as a -integral, see §17.2(v), and more generally Gasper and Rahman (2004, (1.11.2)).
Some of the systems of OP’s that occur in the classification do not have a unique orthogonality property.
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►For other formulas, including -difference equations, recurrence relations, duality formulas, special cases, and limit relations, see Koekoek et al. (2010, Chapter 14).
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3: 2.7 Differential Equations
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►All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients.
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►To include the point at infinity in the foregoing classification scheme, we transform it into the origin by replacing in (2.7.1) with ; see Olver (1997b, pp. 153–154).
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►Hence unless the series (2.7.8) terminate (in which case the corresponding is zero) they diverge.
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►This phenomenon is an example of resurgence, a classification due to Écalle (1981a, b).
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►The exceptional case
is handled by Fabry’s transformation:
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4: 31.12 Confluent Forms of Heun’s Equation
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►This has regular singularities at and , and an irregular singularity of rank 1 at .
►Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions (§30.12) are special cases of solutions of the confluent Heun equation.
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5: 16.4 Argument Unity
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§16.4(i) Classification
… ►The special case is -balanced if is a nonpositive integer and … ►Special cases are as follows: … ►This is (16.4.7) in the case : … ►These series contain symbols as special cases when the parameters are integers; compare §34.4. …6: 16.8 Differential Equations
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