# reflection properties in q

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## 7 matching pages

##### 1: 28.12 Definitions and Basic Properties
28.12.2 $\lambda_{\nu}\left(-q\right)=\lambda_{\nu}\left(q\right)=\lambda_{-\nu}\left(q% \right).$
28.12.15 $\operatorname{se}_{\nu}\left(z,q\right)=-\operatorname{se}_{\nu}\left(-z,q% \right)=-\operatorname{se}_{-\nu}\left(z,q\right).$
##### 2: 28.5 Second Solutions $\operatorname{fe}_{n}$, $\operatorname{ge}_{n}$
$S_{2m+2}(-q)=S_{2m+2}(q).$
##### 3: 28.2 Definitions and Basic Properties
###### Change of Sign of $q$
28.2.37 $\operatorname{se}_{2n+2}\left(z,-q\right)=(-1)^{n}\operatorname{se}_{2n+2}% \left(\tfrac{1}{2}\pi-z,q\right).$
##### 5: 31.8 Solutions via Quadratures
Denote $\mathbf{m}=(m_{0},m_{1},m_{2},m_{3})$ and $\lambda=-4q$. Then …(This $\nu$ is unrelated to the $\nu$ in §31.6.) … The curve $\Gamma$ reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for $m_{j}\in\mathbb{Z}$. When $\lambda=-4q$ approaches the ends of the gaps, the solution (31.8.2) becomes the corresponding Heun polynomial. …
##### 6: 28.31 Equations of Whittaker–Hill and Ince
It has been discussed in detail by Arscott (1967) for $k^{2}<0$, and by Urwin and Arscott (1970) for $k^{2}>0$. … in (28.31.1). … and $m=0,1,\dots,n$ in all cases. … If $p\to\infty$ and $\xi\to 0$ in such a way that $p\xi\to 2q$, then in the notation of §§28.2(v) and 28.2(vi)More important are the double orthogonality relations for $p_{1}\neq p_{2}$ or $m_{1}\neq m_{2}$ or both, given by …
##### 7: Bibliography F
• J. L. Fields and J. Wimp (1961) Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.
• A. S. Fokas and Y. C. Yortsos (1981) The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.
• C. K. Frederickson and P. L. Marston (1992) Transverse cusp diffraction catastrophes produced by the reflection of ultrasonic tone bursts from a curved surface in water. J. Acoust. Soc. Amer. 92 (5), pp. 2869–2877.
• C. K. Frederickson and P. L. Marston (1994) Travel time surface of a transverse cusp caustic produced by reflection of acoustical transients from a curved metal surface. J. Acoust. Soc. Amer. 95 (2), pp. 650–660.
• C. L. Frenzen (1990) Error bounds for a uniform asymptotic expansion of the Legendre function ${Q}^{-m}_{n}({\rm cosh}\,z)$ . SIAM J. Math. Anal. 21 (2), pp. 523–535.