reflection properties in q

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1: 28.12 Definitions and Basic Properties

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28.12.2 $${\lambda }_{\nu }\left(-q\right)={\lambda }_{\nu }\left(q\right)={\lambda }_{-\nu }\left(q\right).$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.12.E2

Encodings:

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See also:

Annotations for §28.12(i), §28.12 and Ch.28

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28.12.10 $$\overline{{\mathrm{me}}_{\nu }(z,q)}={\mathrm{me}}_{\overline{\nu }}(-\overline{z},\overline{q}).$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\overline{z}$: complex conjugate, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.12(ii)

Permalink:

http://dlmf.nist.gov/28.12.E10

Encodings:

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See also:

Annotations for §28.12(ii), §28.12 and Ch.28

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28.12.15 $${\mathrm{se}}_{\nu }(z,q)=-{\mathrm{se}}_{\nu }(-z,q)=-{\mathrm{se}}_{-\nu }(z,q).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.12.E15

Encodings:

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See also:

Annotations for §28.12(iii), §28.12 and Ch.28

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2: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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$$S_{2m+2}(-q)=S_{2m+2}(q).$$

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3: 28.2 Definitions and Basic Properties

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Change of Sign of $q$

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28.2.37 $${\mathrm{se}}_{2n+2}(z,-q)={(-1)}^n{\mathrm{se}}_{2n+2}(\frac{1}{2}\pi -z,q).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $q=h^2$: parameter, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.2.E37

Encodings:

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See also:

Annotations for §28.2(vi), §28.2 and Ch.28

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4: 28.4 Fourier Series

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§28.4(v) Change of Sign of $q$

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5: 31.8 Solutions via Quadratures

… ►Denote $𝐦=(m_0,m_1,m_2,m_3)$ and $\lambda =-4q$. Then …(This $\nu$ is unrelated to the $\nu$ in §31.6.) … ►The curve $\mathrm{\Gamma }$ 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 , and by Urwin and Arscott (1970) for $k^2>0$. … ►in (28.31.1). … ►and $m=0,1,\mathrm{\dots },n$ in all cases. … ►If $p\to \mathrm{\infty }$ 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 $p_1\ne p_2$ or $m_1\ne m_2$ or both, given by …

7: null

error generating summary

8: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►These are orthogonal polynomials for a family of reflection invariant weight functions on the unit sphere. …Let $R_{+}$ be the set of positive roots and let $𝐯\mapsto {\kappa }_{𝐯}$ be a nonnegative function defined on $R_{+}$ with the property that it takes constant value in each conjugacy class of roots. …The weight function is invariant under the reflection group $G$. … ►Let $w_{\kappa }$ be the reflection invariant weight function defined in (37.19.4). … ►Starting from OPs in the $q$-Askey scheme (see §18.27(i)), similar constructions of $d$-variable OPs can be made. …

9: Bibliography F

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Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.

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External Links:

FullText, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.10.

Encodings:

BibTeX

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The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.

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External Links:

ISSN 0024-1318, MathReview (D. A. Lutz)

Referenced by:

§32.10(vi).

Encodings:

BibTeX

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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.

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External Links:

Document

Referenced by:

§36.14(iv).

Encodings:

BibTeX

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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.

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External Links:

Document

Referenced by:

§36.14(iv).

Encodings:

BibTeX

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Error bounds for a uniform asymptotic expansion of the Legendre function $Q_n^{-m}(\cosh z)$ . SIAM J. Math. Anal. 21 (2), pp. 523–535.

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External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.26.

Encodings:

BibTeX

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