# q-differential equations

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##### 2: 17.2 Calculus
$q$-differential equations are considered in §17.6(iv). …
##### 3: 17.13 Integrals
17.13.3 $\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-tq^{\alpha+\beta};q\right)_{\infty}}% {\left(-t;q\right)_{\infty}}\mathrm{d}t=\frac{\Gamma\left(\alpha\right)\Gamma% \left(1-\alpha\right)\Gamma_{q}\left(\beta\right)}{\Gamma_{q}\left(1-\alpha% \right)\Gamma_{q}\left(\alpha+\beta\right)},$
##### 4: 28.12 Definitions and Basic Properties
###### §28.12(i) Eigenvalues $\lambda_{\nu+2n}\left(q\right)$
For given $\nu$ (or $\cos\left(\nu\pi\right)$) and $q$, equation (28.2.16) determines an infinite discrete set of values of $a$, denoted by $\lambda_{\nu+2n}\left(q\right)$, $n=0,\pm 1,\pm 2,\dots$. 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
28.10.9 $\int_{0}^{\ifrac{\pi}{2}}J_{0}\left(2\sqrt{q({\cos}^{2}\tau-{\sin}^{2}\zeta)}% \right)\mathrm{ce}_{2n}\left(\tau,q\right)\mathrm{d}\tau=w_{\mbox{\tiny II}}(% \tfrac{1}{2}\pi;a_{2n}\left(q\right),q)\mathrm{ce}_{2n}\left(\zeta,q\right),$
28.10.10 $\int_{0}^{\pi}J_{0}\left(2\sqrt{q}(\cos\tau+\cos\zeta)\right)\mathrm{ce}_{n}% \left(\tau,q\right)\mathrm{d}\tau=w_{\mbox{\tiny II}}(\pi;a_{n}\left(q\right),% q)\mathrm{ce}_{n}\left(\zeta,q\right).$
##### 6: 28.2 Definitions and Basic Properties
The standard form of Mathieu’s equation with parameters $(a,q)$ is For given $\nu$ 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 $\ifrac{\mathrm{d}}{\mathrm{d}x}$ in the Jacobi, Laguerre, and Hermite cases is played by the $q$-derivative $\mathcal{D}_{q}$, 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). …