# classification of cases

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## 6 matching pages

##### 1: 2.8 Differential Equations with a Parameter

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###### §2.8(i) Classification of Cases

…##### 2: 18.27 $q$-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 $q$-Hahn polynomials and the big $q$-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 $q$-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 $q$-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 $z$ in (2.7.1) with $1/z$; see Olver (1997b, pp. 153–154).
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►Hence unless the series (2.7.8) terminate (in which case the corresponding ${\mathrm{\Lambda}}_{j}$ is zero) they diverge.
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►This phenomenon is an example of

*resurgence*, a classification due to Écalle (1981a, b). … ►The exceptional case ${f}_{0}^{2}=4{g}_{0}$ is handled by*Fabry’s transformation*: …##### 4: 31.12 Confluent Forms of Heun’s Equation

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►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty}$.
►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 ${}_{q+1}{}^{}F_{q}^{}(\mathbf{a};\mathbf{b};1)$ is*$k$-balanced*if ${a}_{q+1}$ is a nonpositive integer and … ►Special cases are as follows: … ►This is (16.4.7) in the case $c=-n$: … ►These series contain $\mathit{6}j$ symbols as special cases when the parameters are integers; compare §34.4. …##### 6: 16.8 Differential Equations

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