Gauss%20multiplication%20formula
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1: 15.5 Derivatives and Contiguous Functions
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§15.5(i) Differentiation Formulas
… βΊThe six functions , , are said to be contiguous to . … βΊ
15.5.12
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βΊBy repeated applications of (15.5.11)–(15.5.18) any function , in which are integers, can be expressed as a linear combination of and any one of its contiguous functions, with coefficients that are rational functions of , and .
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15.5.20
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2: Bibliography K
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On the evaluation of the Gauss hypergeometric function.
C. R. Acad. Bulgare Sci. 45 (6), pp. 35–36.
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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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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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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The Askey scheme as a four-manifold with corners.
Ramanujan J. 20 (3), pp. 409–439.
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3: 27.2 Functions
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§27.2(i) Definitions
… βΊEuclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … βΊGauss and Legendre conjectured that is asymptotic to as : …(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … βΊ
27.2.8
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4: 20 Theta Functions
Chapter 20 Theta Functions
…5: 16.12 Products
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16.12.1
βΊThe following formula is often referred to as Clausen’s formula
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16.12.2
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16.12.3
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6: 20.11 Generalizations and Analogs
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§20.11(i) Gauss Sum
βΊFor relatively prime integers with and even, the Gauss sum is defined by … … βΊ … βΊSimilar identities can be constructed for , , and . …7: 17.7 Special Cases of Higher Functions
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-Analog of Bailey’s Sum
… βΊ-Analog of Gauss’s Sum
… βΊ-Analog of Dixon’s Sum
… βΊGasper–Rahman -Analog of Watson’s Sum
… βΊSecond -Analog of Bailey’s Sum
…8: 15.4 Special Cases
9: Bibliography V
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Multiple Light Scattering.
Vol. 1, Academic Press, New York.
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Some Wonderful Formulas
an Introduction to Polylogarithms.
In Proceedings of the Queen’s Number Theory Conference, 1979
(Kingston, Ont., 1979), R. Ribenboim (Ed.),
Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.
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Certain summation formulae for -series.
J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).
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Transformations of some Gauss hypergeometric functions.
J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
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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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10: Bibliography M
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Rational approximations, software and test methods for sine and cosine integrals.
Numer. Algorithms 12 (3-4), pp. 259–272.
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Hill’s Equation.
Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney.
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Calculation of the modified Bessel functions of the second kind with complex argument.
Math. Comp. 20 (95), pp. 407–412.
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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems.
Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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