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classification of cases


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1: 2.8 Differential Equations with a Parameter
§2.8(i) Classification of Cases
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 Λ 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 = 4 g 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 = . 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
§16.4(i) Classification
The special case F q q + 1 ( 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 6 j symbols as special cases when the parameters are integers; compare §34.4. …
6: 16.8 Differential Equations
§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 F q q + 1 ( 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 α and coalesce into an irregular singularity at . …