Real-rooted atlas

A053117: Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).

A053117

Proved for all n: all roots real, in (−1, 1) · interlacing · basis map preserves real roots what this means

Recurrence from OEIS A053117
\(P_{n}(x) = \left(2\,x\right)\,P_{n-1}(x) + \left(-1\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2022
3-1043
40-4084
510-120165
6060-320326
7-10240-800647
80-80800-19201288

Show all 15 stored rows

Generated b-files: A000027 (row sums), A053117, A053118 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-a2d7d802498a9d22 · revision 1 · JSON
Definition source
oeis_catalog:A053117
Fingerprint
f6-def41fdfbf381d0a

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:  0,  2;
   3: -1,  0,   4;
   4:  0, -4,   0,    8;
   5:  1,  0, -12,    0,  16;
   6:  0,  6,   0,  -32,   0,   32;
   7: -1,  0,  24,    0, -80,    0,   64;
   8:  0, -8,   0,   80,   0, -192,    0,   128;
   9:  1,  0, -40,    0, 240,    0, -448,     0, 256;
  10:  0, 10,   0, -160,   0,  672,    0, -1024,   0, 512;
%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.

A053117: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:  1;
  1:  0,  2;
  2: -1,  0,   4;
  3:  0, -4,   0,    8;
  4:  1,  0, -12,    0,  16;
  5:  0,  6,   0,  -32,   0,   32;
  6: -1,  0,  24,    0, -80,    0,   64;
  7:  0, -8,   0,   80,   0, -192,    0,   128;
  8:  1,  0, -40,    0, 240,    0, -448,     0, 256;
  9:  0, 10,   0, -160,   0,  672,    0, -1024,   0, 512;
%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.

A053118: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:   1;
  1:   2, 0;
  2:   4, 0,    -1;
  3:   8, 0,    -4, 0;
  4:  16, 0,   -12, 0,   1;
  5:  32, 0,   -32, 0,   6, 0;
  6:  64, 0,   -80, 0,  24, 0,   -1;
  7: 128, 0,  -192, 0,  80, 0,   -8, 0;
  8: 256, 0,  -448, 0, 240, 0,  -40, 0,  1;
  9: 512, 0, -1024, 0, 672, 0, -160, 0, 10, 0;
%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-b99a39d00579baec
  • 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-b99a39d00579baec, e-dc563681f97c05c6
  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-1, 1)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-dc563681f97c05c6
  • 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-85cb7c7abd19f98e
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 1] to roots in [-1, 1]: every real polynomial with all roots in E = [-1, 1] has an image with all roots in I = [-1, 1]
    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-85cb7c7abd19f98e

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 (3)

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