expansions%20in%20series%20of%20hypergeometric%20functions
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1: 20.11 Generalizations and Analogs
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§20.11(ii) Ramanujan’s Theta Function and -Series
… ►In the case identities for theta functions become identities in the complex variable , with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►As in §20.11(ii), the modulus of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in -series via (20.9.1). … ►For applications to rapidly convergent expansions for see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004). … ►Multidimensional theta functions with characteristics are defined in §21.2(ii) and their properties are described in §§21.3(ii), 21.5(ii), and 21.6. …2: Bibliography N
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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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Confluent hypergeometric equations and related solvable potentials in quantum mechanics.
J. Math. Phys. 41 (12), pp. 7964–7996.
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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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Diffraction Effects in Semiclassical Scattering.
Montroll Memorial Lecture Series in Mathematical Physics, Cambridge University Press.
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3: Bibliography F
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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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Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series.
Chelsea Publishing Co., New York.
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Asymptotic expansions of the Lebesgue constants for Jacobi series.
Pacific J. Math. 122 (2), pp. 391–415.
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Series expansions of symmetric elliptic integrals.
Math. Comp. 81 (278), pp. 957–990.
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4: 6.20 Approximations
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MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions and , with accuracies up to 20S.
§6.20(ii) Expansions in Chebyshev Series
… ►Luke and Wimp (1963) covers for (20D), and and for (20D).
Luke (1969b, pp. 41–42) gives Chebyshev expansions of , , and for , . The coefficients are given in terms of series of Bessel functions.
Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric -function (§13.2(i)) from which Chebyshev expansions near infinity for , , and follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the functions. If the scheme can be used in backward direction.
5: Bibliography O
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Uniform asymptotic expansions for hypergeometric functions with large parameters. I.
Analysis and Applications (Singapore) 1 (1), pp. 111–120.
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Uniform asymptotic expansions for hypergeometric functions with large parameters. II.
Analysis and Applications (Singapore) 1 (1), pp. 121–128.
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Uniform asymptotic expansions for hypergeometric functions with large parameters. III.
Analysis and Applications (Singapore) 8 (2), pp. 199–210.
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Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms.
SIAM J. Math. Anal. 22 (5), pp. 1475–1489.
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Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations.
Methods Appl. Anal. 1 (1), pp. 1–13.
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6: Bibliography V
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On the series expansion method for computing incomplete elliptic integrals of the first and second kinds.
Math. Comp. 23 (105), pp. 61–69.
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An infinite series of Weber’s parabolic cylinder functions.
Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
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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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Fourier series representation of Ferrers function
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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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7: Bibliography K
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Series expansions for the third incomplete elliptic integral via partial fraction decompositions.
J. Comput. Appl. Math. 207 (2), pp. 331–337.
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Asymptotic expansions of certain -series and a formula of Ramanujan for specific values of the Riemann zeta function.
Acta Arith. 107 (3), pp. 269–298.
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The asymptotic expansion of a hypergeometric function
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Math. Comp. 26 (120), pp. 963.
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Hypergeometric functions of matrix argument are expressible in terms of Appel’s functions
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Proc. Amer. Math. Soc. 70 (1), pp. 39–42.
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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively -binomial sums and basic hypergeometric series.
Séminaire Lotharingien de Combinatoire 30, pp. 61–76.
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8: Bibliography M
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A -analog of the summation theorem for hypergeometric series well-poised in
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Adv. in Math. 57 (1), pp. 14–33.
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A new symmetry related to for classical basic hypergeometric series.
Adv. in Math. 57 (1), pp. 71–90.
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A -analog of hypergeometric series well-poised in
and invariant -functions.
Adv. in Math. 58 (1), pp. 1–60.
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A -analog of the Gauss summation theorem for hypergeometric series in
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Adv. in Math. 72 (1), pp. 59–131.
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A -analog of a Whipple’s transformation for hypergeometric series in
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Adv. Math. 108 (1), pp. 1–76.
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9: Bibliography S
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FGH, a code for the calculation of Coulomb radial wave functions from series expansions.
Comput. Phys. Comm. 146 (2), pp. 250–253.
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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions.
J. Indian Math. Soc. (N. S.) 3, pp. 226–230.
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On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series.
J. Indian Math. Soc. (N. S.) 4, pp. 34–38.
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The expansion of Lamé functions into series of associated Legendre functions of the second kind.
Proc. Cambridge Philos. Soc. 62, pp. 441–452.
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Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
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10: Bibliography D
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Chebyshev series for the spherical Bessel function
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Comput. Phys. Comm. 18 (1), pp. 73–86.
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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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On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function.
Methods Appl. Anal. 5 (3), pp. 223–247.
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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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