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

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2: 18.27 $q$-Hahn Class

… ►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. … ►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. … ►For other formulas, including $q$-difference equations, recurrence relations, duality formulas, special cases, and limit relations, see Koekoek et al. (2010, Chapter 14). …

3: 2.7 Differential Equations

… ►All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … ►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). … ►Hence unless the series (2.7.8) terminate (in which case the corresponding ${\mathrm{\Lambda }}_j$ is zero) they diverge. … ►This phenomenon is an example of resurgence, a classification due to Écalle (1981a, b). … ►The exceptional case $f_0^2=4g_0$ is handled by Fabry’s transformation: …

4: 31.12 Confluent Forms of Heun’s Equation

… ►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. …

5: 16.4 Argument Unity

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§16.4(i) Classification

… ►The special case ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;1)$ 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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§16.8(i) Classification of Singularities

… ►Compare §2.7(i) in the case $n=2$. … ►In each case there are no other singularities. … ►Analytical continuation formulas for ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$ near $z=1$ are given in Bühring (1987b) for the case $q=2$, and in Bühring (1992) for the general case. … ►Thus in the case $p=q$ the regular singularities of the function on the left-hand side at $\alpha$ and $\mathrm{\infty }$ coalesce into an irregular singularity at $\mathrm{\infty }$. …