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reflection properties in q

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1: 28.12 Definitions and Basic Properties
28.12.2 λ ν ( - q ) = λ ν ( q ) = λ - ν ( q ) .
28.12.10 me ν ( z , q ) ¯ = me ν ¯ ( - z ¯ , q ¯ ) .
28.12.15 se ν ( z , q ) = - se ν ( - z , q ) = - se - ν ( z , q ) .
2: 28.5 Second Solutions fe n , ge n
S 2 m + 2 ( - q ) = S 2 m + 2 ( q ) .
3: 28.2 Definitions and Basic Properties
Change of Sign of q
28.2.37 se 2 n + 2 ( z , - q ) = ( - 1 ) n se 2 n + 2 ( 1 2 π - z , q ) .
4: 28.4 Fourier Series
§28.4(v) Change of Sign of q
5: 31.8 Solutions via Quadratures
Denote m = ( m 0 , m 1 , m 2 , m 3 ) and λ = - 4 q . Then …(This ν is unrelated to the ν in §31.6.) … The curve Γ reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for m j . When λ = - 4 q 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 , , n in all cases. … If p and ξ 0 in such a way that p ξ 2 q , then in the notation of §§28.2(v) and 28.2(vi)More important are the double orthogonality relations for p 1 p 2 or m 1 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 n - m ( cosh z ) . SIAM J. Math. Anal. 21 (2), pp. 523–535.