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Stieltjes polynomials

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1: 31.15 Stieltjes Polynomials
§31.15 Stieltjes Polynomials
§31.15(i) Definitions
Stieltjes polynomials are polynomial solutions of the Fuchsian equation (31.14.1). …
§31.15(ii) Zeros
§31.15(iii) Products of Stieltjes Polynomials
2: 18.27 q -Hahn Class
§18.27(vi) Stieltjes–Wigert Polynomials
18.27.18 S n ( x ; q ) = = 0 n q 2 ( - x ) ( q ; q ) ( q ; q ) n - = 1 ( q ; q ) n ϕ 1 1 ( q - n 0 ; q , - q n + 1 x ) .
3: 18.1 Notation
  • Stieltjes–Wigert: S n ( x ; q ) .

  • 4: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
    18.29.2 Q n ( z ; a , b , c , d q ) z n ( a z - 1 , b z - 1 , c z - 1 , d z - 1 ; q ) ( z - 2 , b c , b d , c d ; q ) , n ; z , a , b , c , d , q fixed.
    For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). …
    5: Bibliography V
  • H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.
  • 6: Bibliography W
  • Z. Wang and R. Wong (2006) Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.
  • 7: Bibliography
  • A. M. Al-Rashed and N. Zaheer (1985) Zeros of Stieltjes and Van Vleck polynomials and applications. J. Math. Anal. Appl. 110 (2), pp. 327–339.
  • M. Alam (1979) Zeros of Stieltjes and Van Vleck polynomials. Trans. Amer. Math. Soc. 252, pp. 197–204.
  • 8: Bibliography K
  • E. G. Kalnins and W. Miller (1993) Orthogonal Polynomials on n -spheres: Gegenbauer, Jacobi and Heun. In Topics in Polynomials of One and Several Variables and their Applications, pp. 299–322.
  • T. H. Koornwinder (1977) The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.
  • T. H. Koornwinder (1989) Meixner-Pollaczek polynomials and the Heisenberg algebra. J. Math. Phys. 30 (4), pp. 767–769.
  • T. H. Koornwinder (2012) Askey-Wilson polynomial. Scholarpedia 7 (7), pp. 7761.
  • T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.
  • 9: 25.2 Definition and Expansions
    25.2.4 ζ ( s ) = 1 s - 1 + n = 0 ( - 1 ) n n ! γ n ( s - 1 ) n ,
    where the Stieltjes constants γ n are defined via
    25.2.5 γ n = lim m ( k = 1 m ( ln k ) n k - ( ln m ) n + 1 n + 1 ) .
    25.2.10 ζ ( s ) = 1 s - 1 + 1 2 + k = 1 n ( s + 2 k - 2 2 k - 1 ) B 2 k 2 k - ( s + 2 n 2 n + 1 ) 1 B ~ 2 n + 1 ( x ) x s + 2 n + 1 d x , s > - 2 n , n = 1 , 2 , 3 , .
    For B 2 k see §24.2(i), and for B ~ n ( x ) see §24.2(iii). …
    10: 25.6 Integer Arguments
    §25.6(i) Function Values
    25.6.6 ζ ( 2 k + 1 ) = ( - 1 ) k + 1 ( 2 π ) 2 k + 1 2 ( 2 k + 1 ) ! 0 1 B 2 k + 1 ( t ) cot ( π t ) d t , k = 1 , 2 , 3 , .
    25.6.12 ζ ′′ ( 0 ) = - 1 2 ( ln ( 2 π ) ) 2 + 1 2 γ 2 - 1 24 π 2 + γ 1 ,
    where γ 1 is given by (25.2.5). …