Real-rooted atlas

A053122 → x ↦ x+1

Proved for all n: all roots real, in (−1, 3) · interlacing (weak) what this means

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2-110
30-21-1
411-310
5-123-411
60-426-510
712-9010-61-1
8-139-15-515-710

Show all 15 stored rows

Generated b-files: A011632 (row sums), A104562 ?

Computed in your browser ?

Technical details
Node id
n-2de752bc9bc14cd4 · revision 1 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a6a00d92f59585d4

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

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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2: -1,  1;
   3:  0, -2,   1;
   4:  1,  1,  -3,   1;
   5: -1,  2,   3,  -4,   1;
   6:  0, -4,   2,   6,  -5,   1;
   7:  1,  2,  -9,   0,  10,  -6,   1;
   8: -1,  3,   9, -15,  -5,  15,  -7,  1;
   9:  0, -6,   3,  24, -20, -14,  21, -8,  1;
  10:  1,  3, -18,  -6,  49, -21, -28, 28, -9, 1;
%F Formula
The row polynomials satisfy (2*n-2)*P(n,x) = -(n-n*x)*P(n-1,x) - (3+2*x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - n*x)*P[n - 1, x] - (3 + 2*x - x^2)*D[P[n - 1, x], x])/(2*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(n - n*x)*v[i - 1] - (3 + 2*x - x^2)*deriv(v[i - 1], x))/(2*n - 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)

Generated with polytool c48629374597.

A104562: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:  1;
  1: -1,  1;
  2:  0, -2,   1;
  3:  1,  1,  -3,   1;
  4: -1,  2,   3,  -4,   1;
  5:  0, -4,   2,   6,  -5,   1;
  6:  1,  2,  -9,   0,  10,  -6,   1;
  7: -1,  3,   9, -15,  -5,  15,  -7,  1;
  8:  0, -6,   3,  24, -20, -14,  21, -8,  1;
  9:  1,  3, -18,  -6,  49, -21, -28, 28, -9, 1;
%F Formula
The row polynomials satisfy 2*n*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 >= 1, with P(0,x) = 1.
%t Mathematica
P[0, 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)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (-(n + 1 - (n + 1)*x)*v[i - 1] - (3 + 2*x - x^2)*deriv(v[i - 1], x))/(2*n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 >= 1, is zero or real-rooted with all roots in (-1, 3)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-7ddc15d309ad2a75
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-7ddc15d309ad2a75

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