relations to confluent hypergeometric functions and generalized hypergeometric functions
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31: 13.8 Asymptotic Approximations for Large Parameters
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►For the parabolic cylinder function
see §12.2, and for an extension to an asymptotic expansion see Temme (1978).
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►To obtain approximations for and that hold as , with and combine (13.14.4), (13.14.5) with §13.20(i).
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►For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related results see §13.21(i).
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►For asymptotic approximations to
and as that hold uniformly with respect to
and bounded positive values of , combine (13.14.4), (13.14.5) with §§13.21(ii), 13.21(iii).
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►For generalizations in which is also allowed to be large see Temme and Veling (2022).
32: Bibliography
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Computation of the regular confluent hypergeometric function.
The Mathematica Journal 5 (4), pp. 74–76.
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On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations.
Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.
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Dirichlet series related to the Riemann zeta function.
J. Number Theory 19 (1), pp. 85–102.
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Integral equations and relations for Lamé functions.
Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.
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Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials.
Mem. Amer. Math. Soc. 54 (319), pp. iv+55.
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33: 13.2 Definitions and Basic Properties
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is entire in and , and is a meromorphic function of .
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►Although does not exist when , , many formulas containing continue to apply in their limiting form.
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►In general, has a branch point at .
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►Unless specified otherwise, however, is assumed to have its principal value.
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Kummer’s Transformations
…34: Bibliography Z
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On the Computation of Zeros of Bessel and Bessel-related Functions.
In Proceedings of the Sixth International Colloquium on
Differential Equations (Plovdiv, Bulgaria, 1995), D. Bainov (Ed.),
Utrecht, pp. 409–416.
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The Dilogarithm Function in Geometry and Number Theory.
In Number Theory and Related Topics (Bombay, 1988), R. Askey and others (Eds.),
Tata Inst. Fund. Res. Stud. Math., Vol. 12, pp. 231–249.
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Distribution Theory and Transform Analysis, An Introduction and Generalized Functions with Applications.
Dover, New York.
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Generalized Watson Transforms and Applications to Group Representations.
Ph.D. Thesis, University of Vermont, Burlington,VT.
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Tablitsy vyrozhdennoi gipergeometricheskoi funktsii.
Vyčisl. Centr Akad. Nauk SSSR, Moscow (Russian).
35: 33.23 Methods of Computation
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§33.23(i) Methods for the Confluent Hypergeometric Functions
… ►§33.23(iv) Recurrence Relations
►In a similar manner to §33.23(iii) the recurrence relations of §§33.4 or 33.17 can be used for a range of values of the integer , provided that the recurrence is carried out in a stable direction (§3.6). … ►Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. … ►Noble (2004) obtains double-precision accuracy for for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …36: Bibliography D
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Computational properties of three-term recurrence relations for Kummer functions.
J. Comput. Appl. Math. 233 (6), pp. 1505–1510.
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Bernoulli Numbers and Confluent Hypergeometric Functions.
In Number Theory for the Millennium, I (Urbana, IL, 2000),
pp. 343–363.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions.
Methods Appl. Anal. 6 (3), pp. 21–56.
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Uniform asymptotic expansions for the reverse generalized Bessel polynomials, and related functions.
SIAM J. Math. Anal. 32 (5), pp. 987–1013.
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