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A372267 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 4.

%I A372267 #17 Apr 02 2025 06:47:16
%S A372267 3,3,9,9,8,1,0,4,3,5,8,4,8,5,6,2,6,4,8,0,2,6,6,5,7,5,9,1,0,3,2,4,4,6,
%T A372267 8,7,2,0,0,5,7,5,8,6,9,7,7,0,9,1,4,3,5,2,5,9,2,9,5,3,9,7,6,8,2,1,0,2,
%U A372267 0,0,3,0,4,6,3,2,3,7,0,3,4,4,7,7,8,7,5
%N A372267 Decimal expansion of the smallest positive zero of the Legendre polynomial of degree 4.
%H A372267 Paolo Xausa, <a href="/A372267/b372267.txt">Table of n, a(n) for n = 0..10000</a>
%H A372267 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 A372267 Wikipedia, <a href="https://en.wikipedia.org/wiki/Legendre_polynomials">Legendre polynomials</a>.
%H A372267 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.
%F A372267 Smallest positive root of 35*x^4 - 30*x^2 + 3 = 0.
%F A372267 Equals sqrt((3-2*sqrt(6/5))/7).
%e A372267 0.339981043584856264802665759103244687200575869770914352592953...
%t A372267 First[RealDigits[Root[LegendreP[4, #] &, 3], 10, 100]] (* _Paolo Xausa_, Feb 27 2025 *)
%Y A372267 Cf. A008316, A100258.
%Y A372267 There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
%Y A372267    k | zeros
%Y A372267   ---+--------------------------
%Y A372267    2 | A020760
%Y A372267    3 | A010513/10
%Y A372267    4 | A372267, A372268
%Y A372267    5 | A372269, A372270
%Y A372267    6 | A372271, A372272, A372273
%Y A372267    7 | A372274, A372275, A372276
%K A372267 nonn,cons
%O A372267 0,1
%A A372267 _Pontus von Brömssen_, Apr 25 2024