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Heine transformations (first, second, third)

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1: 1.14 Integral Transforms
§1.14 Integral Transforms
§1.14(i) Fourier Transform
§1.14(iii) Laplace Transform
Fourier Transform
Laplace Transform
2: 14.28 Sums
14.28.1 P ν ( z 1 z 2 - ( z 1 2 - 1 ) 1 / 2 ( z 2 2 - 1 ) 1 / 2 cos ϕ ) = P ν ( z 1 ) P ν ( z 2 ) + 2 m = 1 ( - 1 ) m Γ ( ν - m + 1 ) Γ ( ν + m + 1 ) P ν m ( z 1 ) P ν m ( z 2 ) cos ( m ϕ ) ,
§14.28(ii) Heine’s Formula
14.28.2 n = 0 ( 2 n + 1 ) Q n ( z 1 ) P n ( z 2 ) = 1 z 1 - z 2 , z 1 1 , z 2 2 ,
3: Bibliography V
  • A. L. Van Buren, R. V. Baier, and S. Hanish (1970) A Fortran computer program for calculating the oblate spheroidal radial functions of the first and second kind and their first derivatives. NRL Report No. 6959 Naval Res. Lab.  Washingtion, D.C..
  • A. L. Van Buren and J. E. Boisvert (2004) Improved calculation of prolate spheroidal radial functions of the second kind and their first derivatives. Quart. Appl. Math. 62 (3), pp. 493–507.
  • H. Van de Vel (1969) On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.
  • R. Vidūnas (2005) Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
  • H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.
  • 4: 17.6 ϕ 1 2 Function
    Heine’s First Transformation
    Heine’s Second Tranformation
    Heine’s Third Transformation
    Fine’s First Transformation
    Heine’s Contiguous Relations
    5: 10.1 Special Notation
    The main functions treated in this chapter are the Bessel functions J ν ( z ) , Y ν ( z ) ; Hankel functions H ν ( 1 ) ( z ) , H ν ( 2 ) ( z ) ; modified Bessel functions I ν ( z ) , K ν ( z ) ; spherical Bessel functions j n ( z ) , y n ( z ) , h n ( 1 ) ( z ) , h n ( 2 ) ( z ) ; modified spherical Bessel functions i n ( 1 ) ( z ) , i n ( 2 ) ( z ) , k n ( z ) ; Kelvin functions ber ν ( x ) , bei ν ( x ) , ker ν ( x ) , kei ν ( x ) . … Abramowitz and Stegun (1964): j n ( z ) , y n ( z ) , h n ( 1 ) ( z ) , h n ( 2 ) ( z ) , for j n ( z ) , y n ( z ) , h n ( 1 ) ( z ) , h n ( 2 ) ( z ) , respectively, when n 0 . Jeffreys and Jeffreys (1956): Hs ν ( z ) for H ν ( 1 ) ( z ) , Hi ν ( z ) for H ν ( 2 ) ( z ) , Kh ν ( z ) for ( 2 / π ) K ν ( z ) . Whittaker and Watson (1927): K ν ( z ) for cos ( ν π ) K ν ( z ) . For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
    6: 14.12 Integral Representations
    14.12.2 P ν - μ ( x ) = ( 1 - x 2 ) - μ / 2 Γ ( μ ) x 1 P ν ( t ) ( t - x ) μ - 1 d t , μ > 0 ;
    14.12.3 Q ν μ ( cos θ ) = π 1 / 2 Γ ( ν + μ + 1 ) ( sin θ ) μ 2 μ + 1 Γ ( μ + 1 2 ) Γ ( ν - μ + 1 ) ( 0 ( sinh t ) 2 μ ( cos θ + i sin θ cosh t ) ν + μ + 1 d t + 0 ( sinh t ) 2 μ ( cos θ - i sin θ cosh t ) ν + μ + 1 d t ) , 0 < θ < π , μ > - 1 2 , ν ± μ > - 1 .
    14.12.5 P ν - μ ( x ) = ( x 2 - 1 ) - μ / 2 Γ ( μ ) 1 x P ν ( t ) ( x - t ) μ - 1 d t , μ > 0 .
    14.12.12 Q n m ( x ) = 1 ( n - m ) ! P n m ( x ) x d t ( t 2 - 1 ) ( P n m ( t ) ) 2 , n m .
    Heine’s Integral
    7: 18.11 Relations to Other Functions
    18.11.1 P n m ( x ) = ( 1 2 ) m ( - 2 ) m ( 1 - x 2 ) 1 2 m C n - m ( m + 1 2 ) ( x ) = ( n + 1 ) m ( - 2 ) - m ( 1 - x 2 ) 1 2 m P n - m ( m , m ) ( x ) , 0 m n .
    For the Ferrers function P n m ( x ) , see §14.3(i). …
    §18.11(ii) Formulas of Mehler–Heine Type
    18.11.5 lim n 1 n α P n ( α , β ) ( 1 - z 2 2 n 2 ) = lim n 1 n α P n ( α , β ) ( cos z n ) = 2 α z α J α ( z ) .
    18.11.6 lim n 1 n α L n ( α ) ( z n ) = 1 z 1 2 α J α ( 2 z 1 2 ) .
    8: 19.7 Connection Formulas
    §19.7(i) Complete Integrals of the First and Second Kinds
    Reciprocal-Modulus Transformation
    Imaginary-Modulus Transformation
    Imaginary-Argument Transformation
    For two further transformations of this type see Erdélyi et al. (1953b, p. 316). …
    9: 17.9 Further Transformations of ϕ r r + 1 Functions
    §17.9 Further Transformations of ϕ r r + 1 Functions
    F. H. Jackson’s Transformations
    Transformations of ϕ 2 3 -Series
    Sears–Carlitz Transformation
    Mixed-Base Heine-Type Transformations
    10: 10.42 Zeros
    Properties of the zeros of I ν ( z ) and K ν ( z ) may be deduced from those of J ν ( z ) and H ν ( 1 ) ( z ) , respectively, by application of the transformations (10.27.6) and (10.27.8). For example, if ν is real, then the zeros of I ν ( z ) are all complex unless - 2 < ν < - ( 2 - 1 ) for some positive integer , in which event I ν ( z ) has two real zeros. … K n ( z ) has no zeros in the sector | ph z | 1 2 π ; this result remains true when n is replaced by any real number ν . For the number of zeros of K ν ( z ) in the sector | ph z | π , when ν is real, see Watson (1944, pp. 511–513). For z -zeros of K ν ( z ) , with complex ν , see Ferreira and Sesma (2008). …