homogeneous%20harmonic%20polynomials
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21: Bille C. Carlson
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►In his paper Lauricella’s hypergeometric function
(1963), he defined the -function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter.
…Also, the homogeneity of the -function has led to a new type of mean value for several variables, accompanied by various inequalities.
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22: 36 Integrals with Coalescing Saddles
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23: Gergő Nemes
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►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions.
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24: 33.24 Tables
25: 25.11 Hurwitz Zeta Function
26: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics.
Comput. Phys. Comm. 150 (1), pp. 1–20.
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A harmonic mean inequality for the gamma function.
J. Comput. Appl. Math. 87 (2), pp. 195–198.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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27: 17.17 Physical Applications
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►See Kassel (1995).
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►It involves -generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials.
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28: 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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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
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Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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The constrained quantum mechanical harmonic oscillator.
Proc. Cambridge Philos. Soc. 62, pp. 277–286.
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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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29: 27.15 Chinese Remainder Theorem
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►Their product has 20 digits, twice the number of digits in the data.
…These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result , which is correct to 20 digits.
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30: 2.9 Difference Equations
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►or equivalently the second-order homogeneous linear difference equation
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►This situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)).
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►For applications of asymptotic methods for difference equations to orthogonal polynomials, see, e.
…These methods are particularly useful when the weight function associated with the orthogonal polynomials is not unique or not even known; see, e.
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