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 Functions
§17.9 Further Transformations of Functions
… ►F. H. Jackson’s Transformations
… ►Sears’ Balanced Transformations
… ►Sears–Carlitz Transformation
… ►Mixed-Base Heine-Type Transformations
…3: 16.24 Physical Applications
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►The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner symbols.
These are balanced
functions with unit argument.
Lastly, special cases of the symbols are functions with unit argument.
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4: 16.4 Argument Unity
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Pfaff–Saalschütz Balanced Sum
… ►See Erdélyi et al. (1953a, §4.4(4)) for a non-terminating balanced identity. … ►Balanced series have transformation formulas and three-term relations. … ►Contiguous balanced series have parameters shifted by an integer but still balanced. … ►Transformations for both balanced and very well-poised are included in Bailey (1964, pp. 56–63). …5: 17.4 Basic Hypergeometric Functions
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►
17.4.8
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►The series (17.4.1) is said to be balanced or Saalschützian when it terminates, , , and
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►The series (17.4.1) is said to be k-balanced when and
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6: 18.38 Mathematical Applications
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►It has elegant structures, including -soliton solutions, Lax pairs, and Bäcklund transformations.
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Radon Transform
►See Deans (1983, Chapters 4, 7). … ►See Koornwinder (1981, §4) for details. … ►The symbol (34.4.3), with an alternative expression as a terminating balanced of unit argument, can be expressend in terms of Racah polynomials (18.26.3). …7: Bibliography M
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Associated Wilson polynomials.
Constr. Approx. 7 (4), pp. 521–534.
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Evaluation of complex logarithms and related functions.
SIAM J. Numer. Anal. 18 (4), pp. 744–750.
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Asymptotic approximations for prolate spheroidal wave functions.
Studies in Appl. Math. 54 (4), pp. 315–349.
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A -analog of the summation theorem for hypergeometric series well-poised in
.
Adv. in Math. 57 (1), pp. 14–33.
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Balanced
summation theorems for basic hypergeometric series.
Adv. Math. 131 (1), pp. 93–187.
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8: 16.6 Transformations of Variable
§16.6 Transformations of Variable
►Quadratic
… ►Cubic
►
16.6.2
►For Kummer-type transformations of 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).