Gauss%E2%80%93Laguerre%20formula
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21: 16.16 Transformations of Variables
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§16.16(i) Reduction Formulas
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16.16.1
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16.16.2
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16.16.5
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►See Erdélyi et al. (1953a, §5.10) for these and further reduction formulas.
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22: 27.2 Functions
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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►Gauss and Legendre conjectured that is asymptotic to as :
…(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).)
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23: 18.26 Wilson Class: Continued
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18.26.2
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►See Koekoek et al. (2010, Chapter 9) for further formulas.
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►For the hypergeometric function see §§15.1 and 15.2(i).
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18.26.18
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18.26.19
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24: 15.10 Hypergeometric Differential Equation
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►(b) If equals , and , then fundamental solutions in the neighborhood of are given by and
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§15.10(ii) Kummer’s 24 Solutions and Connection Formulas
… ►The connection formulas for the principal branches of Kummer’s solutions are: …25: 15.3 Graphics
26: 35.9 Applications
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►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument , with and .
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►For other statistical applications of functions of matrix argument see Perlman and Olkin (1980), Groeneboom and Truax (2000), Bhaumik and Sarkar (2002), Richards (2004) (monotonicity of power functions of multivariate statistical test criteria), Bingham et al. (1992) (Procrustes analysis), and Phillips (1986) (exact distributions of statistical test criteria).
These references all use results related to the integral formulas (35.4.7) and (35.5.8).
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27: 8.17 Incomplete Beta Functions
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8.17.7
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8.17.8
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8.17.9
►For the hypergeometric function see §15.2(i).
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8.17.24
positive integers; .
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