Real-rooted atlas

A053122: Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).

A053122

Proved for all n: all roots real, in (0, 4) · interlacing · alternating coefficient signs · basis map preserves real roots what this means

Recurrence from OEIS A053122
\(P_{n}(x) = \left(-2 + x\right)\,P_{n-1}(x) + \left(-1\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = -2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2-21-1
33-410
4-410-611
55-2021-81-1
6-635-5636-1010
77-56126-12055-1211
8-884-252330-22078-141-1

Show all 15 stored rows

Generated b-files: A053122, A053123 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-f7e7f08472ec987a · revision 1 · JSON
Definition source
oeis_catalog:A053122
Fingerprint
f6-63d46ebead6c4d0e

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:  -2,    1;
   3:   3,   -4,    1;
   4:  -4,   10,   -6,    1;
   5:   5,  -20,   21,   -8,     1;
   6:  -6,   35,  -56,   36,   -10,    1;
   7:   7,  -56,  126, -120,    55,  -12,    1;
   8:  -8,   84, -252,  330,  -220,   78,  -14,   1;
   9:   9, -120,  462, -792,   715, -364,  105, -16,   1;
  10: -10,  165, -792, 1716, -2002, 1365, -560, 136, -18, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(2-x)*P(n-1,x) - P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -2 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := -2 + x;
P[n_, x_] := P[n, x] = Expand[-(2 - x)*P[n - 1, x] - P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -2 + x; for(n = 3, N, my(i = n); v[i] = -(2 - x)*v[i - 1] - v[i - 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.

A053122: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:   1;
  1:  -2,    1;
  2:   3,   -4,    1;
  3:  -4,   10,   -6,    1;
  4:   5,  -20,   21,   -8,     1;
  5:  -6,   35,  -56,   36,   -10,    1;
  6:   7,  -56,  126, -120,    55,  -12,    1;
  7:  -8,   84, -252,  330,  -220,   78,  -14,   1;
  8:   9, -120,  462, -792,   715, -364,  105, -16,   1;
  9: -10,  165, -792, 1716, -2002, 1365, -560, 136, -18, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(2-x)*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := -2 + x;
P[n_, x_] := P[n, x] = Expand[-(2 - x)*P[n - 1, x] - P[n - 2, x]];
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; v[2] = -2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(2 - x)*v[i - 1] - v[i - 2]); 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.

A053123: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 1,  -2;
  2: 1,  -4,   3;
  3: 1,  -6,  10,   -4;
  4: 1,  -8,  21,  -20,    5;
  5: 1, -10,  36,  -56,   35,    -6;
  6: 1, -12,  55, -120,  126,   -56,    7;
  7: 1, -14,  78, -220,  330,  -252,   84,   -8;
  8: 1, -16, 105, -364,  715,  -792,  462, -120,   9;
  9: 1, -18, 136, -560, 1365, -2002, 1716, -792, 165, -10;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = -(2-x)*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := -2 + x;
P[n_, x_] := P[n, x] = Expand[-(2 - x)*P[n - 1, x] - P[n - 2, x]];
Table[Reverse[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; v[2] = -2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(2 - x)*v[i - 1] - v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(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 (-inf, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-46262d3eff45ae1d
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-46262d3eff45ae1d, e-8de75f6dc51d73d9
  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (0, 4)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-8de75f6dc51d73d9
  • Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n(-x) have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-8de75f6dc51d73d9
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 1] to roots in (-inf, +inf): every real polynomial with all roots in E = [-1, 1] has a real-rooted image
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-b3a5a732652de064
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 1] to roots in [0, 4]: every real polynomial with all roots in E = [-1, 1] has an image with all roots in I = [0, 4]
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-b3a5a732652de064

Proof chains

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

Used in 3 proof steps

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

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