Real-rooted atlas

A124038 → x ↦ x+1

Proved for all n: all roots real, in (−3, 2) · interlacing (weak) what this means

Recurrence
\(\left(2 - 8\,n + 8\,n^{2}\right)\,P_{n}(x) = \left(1 - 10\,n - 2\,n x + 7\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-3 - 13\,x - 2\,x^{2} - 12\,n + 8\,n x + 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(9 - 6\,x - 3\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1-110
2-201-1
3-1-311-2
41-4-321-3
520-8-231-4
616-5-12041-5
7-179-15-15351-6

Show all 15 stored rows

Generated b-files: A001478 (row sums), A001489 (row sums), A024000 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-366c29e67ccc8d94 · revision 1 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-348de79756a0b070

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 0)
  • 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

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1: -1,  1;
  2: -2,  0,  1;
  3: -1, -3,  1,   1;
  4:  1, -4, -3,   2,   1;
  5:  2,  0, -8,  -2,   3,   1;
  6:  1,  6, -5, -12,   0,   4,   1;
  7: -1,  7,  9, -15, -15,   3,   5,  1;
  8: -2,  0, 21,   6, -30, -16,   7,  6, 1;
  9: -1, -9, 12,  42,  -9, -49, -14, 12, 7, 1;
%F Formula
The row polynomials satisfy (8*n^2-8*n+2)*P(n,x) = (7*n^2-10*n+1+(4*n^2-2*n)*x)*P(n-1,x) - (12*n+3-(8*n-13)*x-(4*n-2)*x^2)*d/dx P(n-1,x) + (9-6*x-3*x^2)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((7*n^2 - 10*n + 1 + (4*n^2 - 2*n)*x)*P[n - 1, x] - (12*n + 3 - (8*n - 13)*x - (4*n - 2)*x^2)*D[P[n - 1, x], x] + (9 - 6*x - 3*x^2)*D[P[n - 1, x], {x, 2}])/(8*n^2 - 8*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((7*n^2 - 10*n + 1 + (4*n^2 - 2*n)*x)*v[i - 1] - (12*n + 3 - (8*n - 13)*x - (4*n - 2)*x^2)*deriv(v[i - 1], x) + (9 - 6*x - 3*x^2)*deriv(deriv(v[i - 1], x), x))/(8*n^2 - 8*n + 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-3, 2)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-480f02718c8257df
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-480f02718c8257df

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023