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A372268 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 4.

%I A372268 #16 Apr 02 2025 06:48:02
%S A372268 8,6,1,1,3,6,3,1,1,5,9,4,0,5,2,5,7,5,2,2,3,9,4,6,4,8,8,8,9,2,8,0,9,5,
%T A372268 0,5,0,9,5,7,2,5,3,7,9,6,2,9,7,1,7,6,3,7,6,1,5,7,2,1,9,2,0,9,0,6,5,2,
%U A372268 9,4,7,1,4,9,5,0,4,8,8,6,5,7,0,4,1,6,2
%N A372268 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 4.
%H A372268 Paolo Xausa, <a href="/A372268/b372268.txt">Table of n, a(n) for n = 0..10000</a>
%H A372268 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 25.4, n=4
%H A372268 Wikipedia, <a href="https://en.wikipedia.org/wiki/Legendre_polynomials">Legendre polynomials</a>.
%H A372268 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.
%F A372268 Largest positive root of 35*x^4 - 30*x^2 + 3 = 0.
%F A372268 Equals sqrt((3+2*sqrt(6/5))/7).
%e A372268 0.861136311594052575223946488892809505095725379629717637615721...
%t A372268 First[RealDigits[Root[LegendreP[4, #] &, 4], 10, 100]] (* _Paolo Xausa_, Feb 27 2025 *)
%Y A372268 Cf. A008316, A100258.
%Y A372268 There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
%Y A372268    k | zeros
%Y A372268   ---+--------------------------
%Y A372268    2 | A020760
%Y A372268    3 | A010513/10
%Y A372268    4 | A372267, A372268
%Y A372268    5 | A372269, A372270
%Y A372268    6 | A372271, A372272, A372273
%Y A372268    7 | A372274, A372275, A372276
%K A372268 nonn,cons
%O A372268 0,1
%A A372268 _Pontus von Brömssen_, Apr 25 2024