chebyshev_t

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

Recurrence

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

\(P_{0}(x) = 1\)

\(P_{1}(x) = x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	1							1
2	-1	0	2						1
3	0	-3	0	4					1
4	1	0	-8	0	8				1
5	0	5	0	-20	0	16			1
6	-1	0	18	0	-48	0	32		1
7	0	-7	0	56	0	-112	0	64	1

Show all 15 stored rows

Generated b-files: A039991 (reversed triangle), A053120 ?

Computed in your browser ?

Technical details

Node id: n-9ae8d431f7b33f22 · revision 1 · JSON
polytool family: chebyshev_t
Definition source: named_family:chebyshev_t
Fingerprint: f6-3d213e09756fa2d2

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 ?

A053120: triangle, 78 terms agree, 12 full rows row-aligned (tabl) · view at the OEIS · generated b-file 990 terms from n = 0 ?
A039991: reversed triangle, 78 terms agree, 12 full rows row-aligned (tabl) in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?

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:  0,  1;
  2: -1,  0,   2;
  3:  0, -3,   0,    4;
  4:  1,  0,  -8,    0,   8;
  5:  0,  5,   0,  -20,   0,   16;
  6: -1,  0,  18,    0, -48,    0,   32;
  7:  0, -7,   0,   56,   0, -112,    0,   64;
  8:  1,  0, -32,    0, 160,    0, -256,    0, 128;
  9:  0,  9,   0, -120,   0,  432,    0, -576,   0, 256;

%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) = x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 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, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[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};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A039991: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:   1;
  1:   1, 0;
  2:   2, 0,   -1;
  3:   4, 0,   -3, 0;
  4:   8, 0,   -8, 0,   1;
  5:  16, 0,  -20, 0,   5, 0;
  6:  32, 0,  -48, 0,  18, 0,   -1;
  7:  64, 0, -112, 0,  56, 0,   -7, 0;
  8: 128, 0, -256, 0, 160, 0,  -32, 0, 1;
  9: 256, 0, -576, 0, 432, 0, -120, 0, 9, 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) = x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 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] = 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.

A053120: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:  1;
  1:  0,  1;
  2: -1,  0,   2;
  3:  0, -3,   0,    4;
  4:  1,  0,  -8,    0,   8;
  5:  0,  5,   0,  -20,   0,   16;
  6: -1,  0,  18,    0, -48,    0,   32;
  7:  0, -7,   0,   56,   0, -112,    0,   64;
  8:  1,  0, -32,    0, 160,    0, -256,    0, 128;
  9:  0,  9,   0, -120,   0,  432,    0, -576,   0, 256;

%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) = x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 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] = 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.

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-efd5e57f56b3429d
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-5437345511af69a3, e-efd5e57f56b3429d
Real-rooted. Every row P_n, n >= 0, 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-5437345511af69a3
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-85b780ace6aa61d5
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-85b780ace6aa61d5

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

basis_map_theorem e-85b780ace6aa61d5 root: theorem check

Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
theorem_check three_term_recurrence e-5437345511af69a3 root: theorem check

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
theorem_check three_term_recurrence e-efd5e57f56b3429d root: theorem check

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Used in 3 proof steps

e-298ee5f2391f30e9 preserver_image (affine, shift 0) → shift_x_plus_one of chebyshev_t
e-525fe5a9d9abc075 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of chebyshev_t
e-c1cd3b615621d834 preserver_image (derivative_polynomial, shift 0) → derivative of chebyshev_t

Related sequences ?

Same fingerprint (1)

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

A039991: Triangle of coefficients of cos(x)^n in polynomial for cos(nx).

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023