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Sears’ balanced 4ϕ3 transformation

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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: 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
SearsBalanced ϕ 3 4 Transformations
Sears–Carlitz Transformation
Mixed-Base Heine-Type Transformations
3: 16.24 Physical Applications
The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner 6 j symbols. These are balanced F 3 4 functions with unit argument. Lastly, special cases of the 9 j symbols are F 4 5 functions with unit argument. …
4: 16.4 Argument Unity
Pfaff–Saalschütz Balanced Sum
See Erdélyi et al. (1953a, §4.4(4)) for a non-terminating balanced identity. … Balanced F 3 4 ( 1 ) series have transformation formulas and three-term relations. … Contiguous balanced series have parameters shifted by an integer but still balanced. … Transformations for both balanced F 3 4 ( 1 ) and very well-poised F 6 7 ( 1 ) are included in Bailey (1964, pp. 56–63). …
5: 17.4 Basic Hypergeometric Functions
17.4.8 Φ ( 4 ) ( a , b ; c , c ; q ; x , y ) = m , n 0 ( a , b ; q ) m + n x m y n ( q , c ; q ) m ( q , c ; q ) n .
The series (17.4.1) is said to be balanced or Saalschützian when it terminates, r = s , z = q , and … The series (17.4.1) is said to be k-balanced when r = s and …
6: 18.38 Mathematical Applications
It has elegant structures, including N -soliton solutions, Lax pairs, and Bäcklund transformations. …
Radon Transform
See Deans (1983, Chapters 4, 7). … See Koornwinder (1981, §4) for details. … The 6 j symbol (34.4.3), with an alternative expression as a terminating balanced F 3 4 of unit argument, can be expressend in terms of Racah polynomials (18.26.3). …
7: Bibliography M
  • D. R. Masson (1991) Associated Wilson polynomials. Constr. Approx. 7 (4), pp. 521–534.
  • G. J. Miel (1981) Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.
  • J. W. Miles (1975) Asymptotic approximations for prolate spheroidal wave functions. Studies in Appl. Math. 54 (4), pp. 315–349.
  • S. C. Milne (1985a) A q -analog of the F 4 5 ( 1 ) summation theorem for hypergeometric series well-poised in 𝑆𝑈 ( n ) . Adv. in Math. 57 (1), pp. 14–33.
  • S. C. Milne (1997) Balanced Θ 2 3 summation theorems for U ( n ) basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.
  • 8: 16.6 Transformations of Variable
    §16.6 Transformations of Variable
    Quadratic
    Cubic
    16.6.2 F 2 3 ( a , 2 b a 1 , 2 2 b + a b , a b + 3 2 ; z 4 ) = ( 1 z ) a F 2 3 ( 1 3 a , 1 3 a + 1 3 , 1 3 a + 2 3 b , a b + 3 2 ; 27 z 4 ( 1 z ) 3 ) .
    For Kummer-type transformations of F 2 2 functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5), Miller and Paris (2011), Choi and Rathie (2013) and Wang and Rathie (2013).
    9: 2.5 Mellin Transform Methods
    §2.5 Mellin Transform Methods
    The Mellin transform of f ( t ) is defined by …
    §2.5(iii) Laplace Transforms with Small Parameters
    For examples in which the integral defining the Mellin transform h ( z ) does not exist for any value of z , see Wong (1989, Chapter 3), Bleistein and Handelsman (1975, Chapter 4), and Handelsman and Lew (1970).
    10: 20.7 Identities
    See also Carlson (2011, §§1 and 4).
    §20.7(iv) Reduction Formulas for Products
    §20.7(vi) Landen Transformations
    §20.7(viii) Transformations of Lattice Parameter
    §20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products