Real-rooted atlas

A053122 → x ↦ x+1 → D

Proved for all n all roots realinterlacing (weak)

Recurrence

\(\left(-4 + 2\,n\right)\,P_{n}(x) = \left(-1 + x - n + n x\right)\,P_{n-1}(x) + \left(-3 - 2\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
3-220
41-63-2
526-1240
6-4418-2053
72-18040-3060
8318-45-2075-427-4

All 15 rows

Technical details
Node id
n-faea90cc70c688ad · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2e9c53d04715c401

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): no
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  0;
   2:  1;
   3: -2,   2;
   4:  1,  -6,   3;
   5:  2,   6, -12,   4;
   6: -4,   4,  18, -20,    5;
   7:  2, -18,   0,  40,  -30,    6;
   8:  3,  18, -45, -20,   75,  -42,   7;
   9: -6,   6,  72, -80,  -70,  126, -56,   8;
  10:  3, -36, -18, 196, -105, -168, 196, -72, 9;
%F Formula
The row polynomials satisfy (2*n-4)*P(n,x) = -(n+1-(n+1)*x)*P(n-1,x) - (3+2*x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 - (n + 1)*x)*P[n - 1, x] - (3 + 2*x - x^2)*D[P[n - 1, x], x])/(2*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(n + 1 - (n + 1)*x)*v[i - 1] - (3 + 2*x - x^2)*deriv(v[i - 1], x))/(2*n - 4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023