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A384588 Decimal expansion of the weight factor for Laguerre-Gauss quadrature corresponding to abscissa A384586.

%I A384588 #11 Jun 26 2025 08:00:59
%S A384588 0,3,8,8,8,7,9,0,8,5,1,5,0,0,5,3,8,4,2,7,2,4,3,8,1,6,8,1,5,6,2,0,9,9,
%T A384588 1,3,7,2,2,3,0,7,1,9,1,3,4,8,2,7,6,9,0,2,1,8,1,6,3,5,2,9,2,4,0,4,5,2,
%U A384588 5,7,6,2,9,1,0,1,7,6,9,8,0,9,9,9,8,4,3,3
%N A384588 Decimal expansion of the weight factor for Laguerre-Gauss quadrature corresponding to abscissa A384586.
%H A384588 Paolo Xausa, <a href="/A384588/b384588.txt">Table of n, a(n) for n = 0..10000</a>
%H A384588 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.9, n = 4.
%H A384588 A.H.M. Smeets, <a href="/A384277/a384277.txt">Abscissas and weight factors for Laguerre integration for some larger degrees</a>.
%H A384588 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Laguerre-GaussQuadrature.html">Laguerre-Gauss Quadrature</a>.
%H A384588 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.
%F A384588 Second smallest root of 1990656*x^4 - 1990656*x^3 + 504576*x^2 - 16960*x + 9 = 0.
%e A384588 0.038887908515005384272438168156209913722307191348276...
%t A384588 First[RealDigits[Root[1990656*#^4 - 1990656*#^3 + 504576*#^2 - 16960*# + 9 &, 2], 10, 100, -1]] (* _Paolo Xausa_, Jun 26 2025 *)
%o A384588 (PARI) solve(x = 0.1, 0.04, 1990656*x^4 - 1990656*x^3 + 504576*x^2 - 16960*x + 9)
%Y A384588 There are k positive real zeros of the Laguerre polynomial of degree k:
%Y A384588    k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
%Y A384588   ---+------------------------------------------+-----------------------------------------------------
%Y A384588    2 | A101465, 1+A014176                       | A201488, A100954-3
%Y A384588    3 | A384277, A384278, A384279                | A384463, A384464, A384465
%Y A384588    4 | A384280, A384281, A384586, A384587       | A384466, A384467, this sequence, A384589
%K A384588 nonn,cons
%O A384588 0,2
%A A384588 _A.H.M. Smeets_, Jun 07 2025