F. H. Jackson transformations
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1: 1.14 Integral Transforms
§1.14 Integral Transforms
… โบIn this subsection we let . โบIf is absolutely integrable on , then is continuous, as , and … โบIf and are absolutely integrable on , then so is , and its Fourier transform is , where is the Fourier transform of . … โบIn this subsection we let , , , and . …2: 31.7 Relations to Other Functions
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โบOther reductions of to a , with at least one free parameter, exist iff the pair takes one of a finite number of values, where .
…They are analogous to quadratic and cubic hypergeometric transformations (§§15.8(iii)–15.8(v)).
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31.7.2
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โบJoyce (1994) gives a reduction in which the independent variable is transformed not polynomially or rationally, but algebraically.
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โบThe solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively.
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3: 33.2 Definitions and Basic Properties
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§33.2(ii) Regular Solution
โบThe function is recessive (§2.7(iii)) at , and is defined by … โบ§33.2(iii) Irregular Solutions
… โบ and are complex conjugates, and their real and imaginary parts are given by … โบAs in the case of , the solutions and are analytic functions of when . …4: 35.8 Generalized Hypergeometric Functions of Matrix Argument
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§35.8(iii) Case
โบKummer Transformation
… โบThomae Transformation
… โบMultidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions and of matrix argument. A similar result for the function of matrix argument is given in Faraut and Korányi (1994, p. 346). …5: 16.4 Argument Unity
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โบThe function is well-poised if
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โบThe function with argument unity and general values of the parameters is discussed in Bühring (1992).
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โบFor generalizations involving functions see Kim et al. (2013).
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โบBalanced series have transformation formulas and three-term relations.
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โบTransformations for both balanced and very well-poised are included in Bailey (1964, pp. 56–63).
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6: 16.6 Transformations of Variable
§16.6 Transformations of Variable
โบQuadratic
… โบCubic
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16.6.2
โบFor 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).
7: 33.8 Continued Fractions
8: Bibliography
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Efficient approximation of the Struve functions occurring in the calculation of sound radiation quantities.
The Journal of the Acoustical Society of America 140 (6), pp. 4154–4160.
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Lectures on Integral Transforms.
Translations of Mathematical Monographs, Vol. 70, American Mathematical Society, Providence, RI.
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Shanks’ convergence acceleration transform, Padé approximants and partitions.
J. Combin. Theory Ser. A 43 (1), pp. 70–84.
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Special value of the hypergeometric function and connection formulae among asymptotic expansions.
J. Indian Math. Soc. (N.S.) 51, pp. 161–221.
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Derivatives and integrals with respect to the order of the Struve functions and
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J. Math. Anal. Appl. 137 (1), pp. 17–36.
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