A372275 Decimal expansion of the middle positive zero of the Legendre polynomial of degree 7.

7, 4, 1, 5, 3, 1, 1, 8, 5, 5, 9, 9, 3, 9, 4, 4, 3, 9, 8, 6, 3, 8, 6, 4, 7, 7, 3, 2, 8, 0, 7, 8, 8, 4, 0, 7, 0, 7, 4, 1, 4, 7, 6, 4, 7, 1, 4, 1, 3, 9, 0, 2, 6, 0, 1, 1, 9, 9, 5, 5, 3, 5, 1, 9, 6, 7, 4, 2, 9, 8, 7, 4, 6, 7, 2, 1, 8, 0, 5, 1, 3, 7, 9, 2, 8, 2, 6

Offset

0,1

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.4, n=7.

A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.

Eric Weisstein's World of Mathematics, Legendre Polynomial.

Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Wikipedia, Legendre polynomials.

Index entries for algebraic numbers, degree 6.

Formula

Middle positive root of 429*x^6 - 693*x^4 + 315*x^2 - 35 = 0.

Example

0.741531185599394439863864773280788407074147647141390260119955...

Mathematica

First[RealDigits[Root[LegendreP[7, #] &, 6], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

Prog

(PARI) solve (x = 0.6, 0.8, 429*x^6 - 693*x^4 + 315*x^2 - 35) \\ A.H.M. Smeets, May 31 2025

Crossrefs

Cf. A008316, A100258.

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:

k | zeros | corresponding weights for Legendre-Gauss quadrature

---+---------------------------------+----------------------------------------------------

2 | A020760 | A000007*10

3 | A010513/10 | A010716

4 | A372267, A372268 | A382103, A382104

5 | A372269, A372270 | A382105, A382106

6 | A372271, A372272, A372273 | A382107, A382686, A382687

7 | A372274, this sequence, A372276 | A382688, A382689, A382690

Sequence in context: A199157 A376845 A328416 * A373809 A010508 A384563

Adjacent sequences: A372272 A372273 A372274 * A372276 A372277 A372278

Keyword

nonn,cons

Author

Pontus von Brömssen, Apr 25 2024

Status

approved