Real-rooted atlas

chebyshev_t → D

Proved for all n: all roots real · interlacing (weak) what this means

Recurrence
\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(n^{2} x\right)\,P_{n-1}(x) + \left(-n + n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2044
3-30129
40-1603216
550-6008025
60360-192019236
7-701680-560044849

Show all 15 stored rows

Generated b-files: A000290 (row sums), A136160 ?

Computed in your browser ?

Technical details
Node id
n-6ee7e6c62309a510 · revision 1 · JSON
Definition source
derived:derivative
Fingerprint
f6-def41fdfbf381d0a

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  0;
  1:  1;
  2:  0,   4;
  3: -3,   0,   12;
  4:  0, -16,    0,   32;
  5:  5,   0,  -60,    0,   80;
  6:  0,  36,    0, -192,    0,   192;
  7: -7,   0,  168,    0, -560,     0,   448;
  8:  0, -64,    0,  640,    0, -1536,     0, 1024;
  9:  9,   0, -360,    0, 2160,     0, -4032,    0, 2304;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = n^2*x*P(n-1,x) - (n-n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(n^2*x*P[n - 1, x] - (n - n*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (n^2*x*v[i - 1] - (n - n*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); 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.

A136160: triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1:  1;
   2:  0,   4;
   3: -3,   0,   12;
   4:  0, -16,    0,    32;
   5:  5,   0,  -60,     0,   80;
   6:  0,  36,    0,  -192,    0,   192;
   7: -7,   0,  168,     0, -560,     0,   448;
   8:  0, -64,    0,   640,    0, -1536,     0,   1024;
   9:  9,   0, -360,     0, 2160,     0, -4032,      0, 2304;
  10:  0, 100,    0, -1600,    0,  6720,     0, -10240,    0, 5120;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = n^2*x*P(n-1,x) - (n-n*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^2*x*P[n - 1, x] - (n - n*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {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^2*x*v[i - 1] - (n - n*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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-c1cd3b615621d834
  • 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-c1cd3b615621d834

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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023