A393368 Decimal expansion of the smallest positive zero of the Hermite polynomial of degree 7.

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

Offset

0,1

Comments

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

k | zeros | corresponding weights for Hermite-Gauss quadrature

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

2 | A010503 | A019704

3 | 0, A115754 | 10*A019717, A019708

4 | A393353, A393354 | A393355, A393356

5 | 0, A393357, A393358 | A245887, A393360, A393361

6 | A393362, A393363, A393364 | A393365, A393366, A393367

7 | 0, this sequence, A393369, A393370 | A393371, A393372, A393373, A393374

Links

Table of n, a(n) for n=0..87.

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]. Eq. 25.4.46 p. 890 and Table 25.10 p. 924.

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

Index entries for algebraic numbers, degree 6.

Formula

Minimal polynomial 8*x^6 -84*x^4 +210*x^2 -105. - R. J. Mathar, Jun 03 2026

Example

0.816287882858964663038710959027145816742889...

Mathematica

RealDigits[x /. FindRoot[HermiteH[7, x], {x, 1}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Mar 02 2026 *)

Prog

(PARI) polrootsreal(polhermite(7))[5]

Crossrefs

Cf. A060821.

Sequence in context: A154213 A353920 A363923 * A176456 A351211 A033812

Adjacent sequences: A393365 A393366 A393367 * A393369 A393370 A393371

Keyword

nonn,cons

Author

A.H.M. Smeets, Mar 02 2026

Status

approved