duplication%20formulas
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21: 3.8 Nonlinear Equations
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§3.8(iv) Zeros of Polynomials
… ►Explicit formulas for the zeros are available if ; see §§1.11(iii) and 4.43. No explicit general formulas exist when . … ►Consider and . We have and . …22: Bibliography G
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Contiguous relations and summation and transformation formulae for basic hypergeometric series.
J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
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Construction of Gauss-Christoffel quadrature formulas.
Math. Comp. 22, pp. 251–270.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Explicit formulas for Bernoulli numbers.
Amer. Math. Monthly 79, pp. 44–51.
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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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23: Bibliography I
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The real roots of Bernoulli polynomials.
Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent.
Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).
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Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution.
J. Phys. A 31 (17), pp. 4073–4113.
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24: 18.5 Explicit Representations
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§18.5(ii) Rodrigues Formulas
… ►Related formula: …See (Erdélyi et al., 1953b, §10.9(37)) for a related formula for ultraspherical polynomials. … ►and two similar formulas by symmetry; compare the second row in Table 18.6.1. … ►For corresponding formulas for Chebyshev, Legendre, and the Hermite polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). …25: Bibliography O
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Fourier Expansions. A Collection of Formulas.
Academic Press, New York-London.
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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series.
J. Inst. Math. Appl. 20 (3), pp. 379–391.
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Connection formulas for second-order differential equations with multiple turning points.
SIAM J. Math. Anal. 8 (1), pp. 127–154.
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Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities.
SIAM J. Math. Anal. 8 (4), pp. 673–700.
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General connection formulae for Liouville-Green approximations in the complex plane.
Philos. Trans. Roy. Soc. London Ser. A 289, pp. 501–548.
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26: 8 Incomplete Gamma and Related
Functions
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27: 28 Mathieu Functions and Hill’s Equation
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28: 25.6 Integer Arguments
29: Bibliography W
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Reduction formulae for products of theta functions.
J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.
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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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Recursion formulae for hypergeometric functions.
Math. Comp. 22 (102), pp. 363–373.
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Explicit formulas for the associated Jacobi polynomials and some applications.
Canad. J. Math. 39 (4), pp. 983–1000.
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Quadrature formulas for oscillatory integral transforms.
Numer. Math. 39 (3), pp. 351–360.
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