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q-differential equations

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1: 17.6 ϕ 1 2 Function
§17.6(iv) Differential Equations
q -Differential Equation
2: 17.2 Calculus
q -differential equations are considered in §17.6(iv). …
3: 17.13 Integrals
17.13.3 0 t α - 1 ( - t q α + β ; q ) ( - t ; q ) d t = Γ ( α ) Γ ( 1 - α ) Γ q ( β ) Γ q ( 1 - α ) Γ q ( α + β ) ,
4: 28.12 Definitions and Basic Properties
§28.12(i) Eigenvalues λ ν + 2 n ( q )
For given ν (or cos ( ν π ) ) and q , equation (28.2.16) determines an infinite discrete set of values of a , denoted by λ ν + 2 n ( q ) , n = 0 , ± 1 , ± 2 , . When q = 0 Equation (28.2.16) has simple roots, given by … … If q is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of z and q by the normalization …
5: 28.10 Integral Equations
6: 28.2 Definitions and Basic Properties
The standard form of Mathieu’s equation with parameters ( a , q ) is For given ν and q , equation (28.2.16) determines an infinite discrete set of values of a , the eigenvalues or characteristic values, of Mathieu’s equation. …
Change of Sign of q
For simple roots q of the corresponding equations (28.2.21) and (28.2.22), the functions are made unique by the normalizations …
7: 18.27 q -Hahn Class
In the q -Hahn class OP’s the role of the operator d / d x in the Jacobi, Laguerre, and Hermite cases is played by the q -derivative 𝒟 q , as defined in (17.2.41). … For other formulas, including q -difference equations, recurrence relations, duality formulas, special cases, and limit relations, see Koekoek et al. (2010, Chapter 14). …
§18.27(ii) q -Hahn Polynomials
§18.27(iii) Big q -Jacobi Polynomials
Discrete q -Hermite I