Real-rooted atlas

A267482: Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=n.

A267482

Proved for all n: all roots real, in (−2, 2) · interlacing · basis map preserves real roots what this means

Recurrence from OEIS A267482
\(P_{n}(x) = \left(x\right)\,P_{n-1}(x) + \left(-1\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
2-1111
3-1-211-1
41-2-311-2
513-3-411-1
6-136-4-5111
7-1-4610-5-6112
81-4-101015-6-7111
915-10-201521-7-811-1
10-1515-20-352128-8-911-2
11-1-61535-35-562836-9-1011-1
121-6-213570-56-843645-10-11111
1317-21-5670126-84-1204555-11-12112
14-1728-56-126126210-120-1655566-12-13111

Generated b-files: A057079 (row sums), A087204 (row sums), A267482 ?

Computed in your browser ?

Technical details
Node id
n-0b318b4ec2091cb2 · revision 1 · JSON
Definition source
oeis_catalog:A267482
Fingerprint
f6-5cfd046d7539c83b

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: -1,  1,   1;
  3: -1, -2,   1,   1;
  4:  1, -2,  -3,   1,  1;
  5:  1,  3,  -3,  -4,  1,  1;
  6: -1,  3,   6,  -4, -5,  1,  1;
  7: -1, -4,   6,  10, -5, -6,  1,  1;
  8:  1, -4, -10,  10, 15, -6, -7,  1, 1;
  9:  1,  5, -10, -20, 15, 21, -7, -8, 1, 1;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] - P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = 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 - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A267482: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:  1;
  1:  1,  1;
  2: -1,  1,   1;
  3: -1, -2,   1,   1;
  4:  1, -2,  -3,   1,  1;
  5:  1,  3,  -3,  -4,  1,  1;
  6: -1,  3,   6,  -4, -5,  1,  1;
  7: -1, -4,   6,  10, -5, -6,  1,  1;
  8:  1, -4, -10,  10, 15, -6, -7,  1, 1;
  9:  1,  5, -10, -20, 15, 21, -7, -8, 1, 1;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = 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.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, 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-d1dad2b285ccdcec
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-d1dad2b285ccdcec, e-d92e4a9b9719ac57
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 2)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-d92e4a9b9719ac57
  • 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-23f2fa3b69d7e23d
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 1] to roots in [-2, 2]: every real polynomial with all roots in E = [-1, 1] has an image with all roots in I = [-2, 2]
    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-23f2fa3b69d7e23d

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

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