reflection properties in q
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9 matching pages
1: 28.12 Definitions and Basic Properties
2: 28.5 Second Solutions ,
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3: 28.2 Definitions and Basic Properties
4: 28.4 Fourier Series
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§28.4(v) Change of Sign of
…5: 31.8 Solutions via Quadratures
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►Denote and .
Then
…(This is unrelated to the
in §31.6.)
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►The curve
reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for .
When approaches the ends of the gaps, the solution (31.8.2) becomes the corresponding Heun polynomial.
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6: 28.31 Equations of Whittaker–Hill and Ince
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►It has been discussed in detail by Arscott (1967) for , and by Urwin and Arscott (1970) for .
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►in (28.31.1).
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►and
in all cases.
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►If and
in such a way that , then in the notation of §§28.2(v) and 28.2(vi)
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►More important are the double orthogonality relations for or or both, given by
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7: null
error generating summary8: 37.19 Other Orthogonal Polynomials of Variables
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►These are orthogonal polynomials for a family of reflection invariant weight functions on the unit sphere.
…Let be the set of positive roots and let be a nonnegative function defined on with the property that it takes constant value in each conjugacy class of roots.
…The weight function is invariant under the reflection group .
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►Let be the reflection invariant weight function defined in (37.19.4).
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►Starting from OPs in the -Askey scheme (see §18.27(i)), similar constructions of -variable OPs can be made.
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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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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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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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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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Error bounds for a uniform asymptotic expansion of the Legendre function
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SIAM J. Math. Anal. 21 (2), pp. 523–535.
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