generalized%20hypergeometric%20functions
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1: Bibliography K
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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On the zeros of some generalized hypergeometric functions.
J. Math. Anal. Appl. 243 (2), pp. 249–260.
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Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters.
J. B. Wolters, Groningen.
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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators.
SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.
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Some special cases of the generalized hypergeometric function
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J. Comput. Appl. Math. 78 (1), pp. 79–95.
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2: Software Index
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►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support.
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►In the list below we identify four main sources of software for computing special functions.
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Commercial Software.
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►The following are web-based software repositories with significant holdings in the area of special functions.
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Open Source | With Book | Commercial | |||||||||||||||||||||||
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20 Theta Functions | |||||||||||||||||||||||||
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Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.
3: Bibliography V
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Transformations of some Gauss hypergeometric functions.
J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
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Generalized Associated Legendre Functions and their Applications.
World Scientific Publishing Co. Inc., Singapore.
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A note on the asymptotic expansion of generalized hypergeometric functions.
Anal. Appl. (Singap.) 12 (1), pp. 107–115.
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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation.
Constr. Approx. 20 (1), pp. 39–54.
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Asymptotic expansion of the generalized hypergeometric function
as for
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Anal. Appl. (Singap.) 21 (2), pp. 535–545.
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4: Bibliography
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Computation of the regular confluent hypergeometric function.
The Mathematica Journal 5 (4), pp. 74–76.
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Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions.
J. Heat Transfer 118 (3), pp. 789–792.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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Applications of basic hypergeometric functions.
SIAM Rev. 16 (4), pp. 441–484.
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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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5: Bibliography B
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Products of generalized hypergeometric series.
Proc. London Math. Soc. (2) 28 (2), pp. 242–254.
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Transformations of generalized hypergeometric series.
Proc. London Math. Soc. (2) 29 (2), pp. 495–502.
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Generalized Hypergeometric Series.
Stechert-Hafner, Inc., New York.
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An analytic continuation formula for the generalized hypergeometric function.
SIAM J. Math. Anal. 19 (5), pp. 1249–1251.
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Generalized hypergeometric functions at unit argument.
Proc. Amer. Math. Soc. 114 (1), pp. 145–153.
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6: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann.
Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).
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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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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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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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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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7: Bibliography N
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Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes.
ACM Trans. Math. Software 18 (3), pp. 345–349.
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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Generalization of Binet’s Gamma function formulas.
Integral Transforms Spec. Funct. 24 (8), pp. 597–606.
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A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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Hypergeometric functions.
Acta Math. 94, pp. 289–349.
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8: Bibliography C
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Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams.
J. Comput. Appl. Math. 115 (1-2), pp. 93–99.
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Generalized incomplete gamma functions with applications.
J. Comput. Appl. Math. 55 (1), pp. 99–124.
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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On zeros of the hypergeometric function.
Serdica 7 (3), pp. 243–249.
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9: Bibliography F
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Generalized parabolic cylinder functions.
Asymptotic Anal. 5 (6), pp. 517–531.
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Uniform asymptotic expansions for hypergeometric functions with large parameters IV.
Anal. Appl. (Singap.) 12 (6), pp. 667–710.
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Expansions of hypergeometric functions in hypergeometric functions.
Math. Comp. 15 (76), pp. 390–395.
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Computing the hypergeometric function.
J. Comput. Phys. 137 (1), pp. 79–100.
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10: Bibliography W
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Infinitely differentiable generalized logarithmic and exponential functions.
Math. Comp. 57 (196), pp. 723–733.
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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Some transformations of generalized hypergeometric series.
Proc. London Math. Soc. (2) 26 (2), pp. 257–272.
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The asymptotic expansion of the generalized Bessel function.
Proc. London Math. Soc. (2) 38, pp. 257–270.
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The asymptotic expansion of the generalized hypergeometric function.
Proc. London Math. Soc. (2) 46, pp. 389–408.
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