Real-rooted atlas

A110162 → D+1

Proved for all n all roots realinterlacing (weak)

Recurrence

\(\left(-8 - 20\,x + 4\,x^{2} + 20\,n + 10\,n x - 2\,n x^{2} - 12\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(-4 + 14\,x - 12\,x^{2} + 2\,x^{3} - 6\,n - 3\,n x + 6\,n x^{2} - n x^{3} + 8\,n^{2} - 4\,n^{2} x - 2\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-16\,x - 8\,x^{2} + 7\,x^{3} - x^{4} + 12\,n x - 3\,n x^{2} - 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = -1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2-110
3-2-21-3
47-3-312
5-1424-4-413
623-7555-5-51-6
7-34174-231104-6-612
847-343686-546175-7-716

All 15 rows

Technical details
Node id
n-0722f6e57948ddf4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8ba314666b9a1402

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • 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:   1;
   2:  -1,    1;
   3:  -2,   -2,     1;
   4:   7,   -3,    -3,     1;
   5: -14,   24,    -4,    -4,     1;
   6:  23,  -75,    55,    -5,    -5,     1;
   7: -34,  174,  -231,   104,    -6,    -6,   1;
   8:  47, -343,   686,  -546,   175,    -7,  -7,  1;
   9: -62,  608, -1680,  1968, -1100,   272,  -8, -8,  1;
  10:  79, -999,  3618, -5742,  4653, -1989, 399, -9, -9, 1;
%F Formula
The row polynomials satisfy (2*n^3-12*n^2+20*n-8+(10*n-20)*x-(2*n-4)*x^2)*P(n,x) = -(2*n^3-8*n^2+6*n+4-(n^3-4*n^2-3*n+14)*x-(6*n-12)*x^2+(n-2)*x^3)*P(n-1,x) - ((4*n^2-12*n+16)*x-(n^2-3*n-8)*x^2-7*x^3+x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = -1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := -1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 8*n^2 + 6*n + 4 - (n^3 - 4*n^2 - 3*n + 14)*x - (6*n - 12)*x^2 + (n - 2)*x^3)*P[n - 1, x] - ((4*n^2 - 12*n + 16)*x - (n^2 - 3*n - 8)*x^2 - 7*x^3 + x^4)*D[P[n - 1, x], x])/(2*n^3 - 12*n^2 + 20*n - 8 + (10*n - 20)*x - (2*n - 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -1 + x; for(n = 3, N, my(i = n); v[i] = (-(2*n^3 - 8*n^2 + 6*n + 4 - (n^3 - 4*n^2 - 3*n + 14)*x - (6*n - 12)*x^2 + (n - 2)*x^3)*v[i - 1] - ((4*n^2 - 12*n + 16)*x - (n^2 - 3*n - 8)*x^2 - 7*x^3 + x^4)*deriv(v[i - 1], x))/(2*n^3 - 12*n^2 + 20*n - 8 + (10*n - 20)*x - (2*n - 4)*x^2)); 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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023