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Neumann polynomial

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1: 10.23 Sums
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10.23.11 a k = 1 2 ⁢ π ⁢ i ⁢ | t | = c f ⁡ ( t ) ⁢ O k ⁡ ( t ) ⁢ d t , 0 < c < c ,
►and O k ⁡ ( t ) is Neumann’s polynomial, defined by the generating function: ►
10.23.12 1 t z = J 0 ⁡ ( z ) ⁢ O 0 ⁡ ( t ) + 2 ⁢ k = 1 J k ⁡ ( z ) ⁢ O k ⁡ ( t ) , | z | < | t | .
► O n ⁡ ( t ) is a polynomial of degree n + 1 in 1 / t : O 0 ⁡ ( t ) = 1 / t and ►
10.23.13 O n ⁡ ( t ) = 1 4 ⁢ k = 0 n / 2 ( n k 1 ) ! ⁢ n k ! ⁢ ( 2 t ) n 2 ⁢ k + 1 , n = 1 , 2 , .
2: Errata
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  • Equation (10.23.11)
    10.23.11 a k = 1 2 ⁢ π ⁢ i ⁢ | t | = c f ⁢ ( t ) ⁢ O k ⁡ ( t ) ⁢ d t , 0 < c < c

    Originally the contour of integration written incorrectly as | z | = c , has been corrected to be | t | = c .

    Reported by Mark Dunster on 2021-03-22

  • 3: 14.12 Integral Representations
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    14.12.5 P ν μ ⁡ ( x ) = ( x 2 1 ) μ / 2 Γ ⁡ ( μ ) ⁢ 1 x P ν ⁡ ( t ) ⁢ ( x t ) μ 1 ⁢ d t , ⁡ μ > 0 .
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    Neumann’s Integral
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    14.12.13 Q n ⁡ ( x ) = 1 2 ⁢ ( n ! ) ⁢ 1 1 P n ⁡ ( t ) x t ⁢ d t .
    4: Bibliography S
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  • H. E. Salzer (1955) Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.
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  • I. Sh. SlavutskiÄ­ (1995) Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.
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  • W. F. Sun (1996) Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.
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  • O. Szász (1950) On the relative extrema of ultraspherical polynomials. Boll. Un. Mat. Ital. (3) 5, pp. 125–127.
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  • G. Szegő (1967) Orthogonal Polynomials. 3rd edition, American Mathematical Society, New York.