chebyshev_t → D+1

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

Recurrence

\(\left(-2 - x + x^{2} + 4\,n + n x - n x^{2} - 3\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(-1 - x + x^{3} + n + n x - n x^{3} - n^{2} x + n^{3} x\right)\,P_{n-1}(x) + \left(-1 + x + 2\,x^{2} - x^{3} - x^{4} + n - n x^{2} - n^{2} + n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	-1	4	2													5
3	-3	-3	12	4												10
4	1	-16	-8	32	8											17
5	5	5	-60	-20	80	16										26
6	-1	36	18	-192	-48	192	32									37
7	-7	-7	168	56	-560	-112	448	64								50
8	1	-64	-32	640	160	-1536	-256	1024	128							65
9	9	9	-360	-120	2160	432	-4032	-576	2304	256						82
10	-1	100	50	-1600	-400	6720	1120	-10240	-1280	5120	512					101
11	-11	-11	660	220	-6160	-1232	19712	2816	-25344	-2816	11264	1024				122
12	1	-144	-72	3360	840	-21504	-3584	55296	6912	-61440	-6144	24576	2048			145
13	13	13	-1092	-364	14560	2912	-69888	-9984	149760	16640	-146432	-13312	53248	4096		170
14	-1	196	98	-6272	-1568	56448	9408	-215040	-26880	394240	39424	-344064	-28672	114688	8192	197

Generated b-files: A002522 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-4408f155e5aec2f8 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-7449cf8241acaca5

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 ?

A002522: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
A160457: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A217990: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A098749: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A098748: row sums, 12 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,   4,    2;
  3: -3,  -3,   12,    4;
  4:  1, -16,   -8,   32,    8;
  5:  5,   5,  -60,  -20,   80,    16;
  6: -1,  36,   18, -192,  -48,   192,    32;
  7: -7,  -7,  168,   56, -560,  -112,   448,   64;
  8:  1, -64,  -32,  640,  160, -1536,  -256, 1024,  128;
  9:  9,   9, -360, -120, 2160,   432, -4032, -576, 2304, 256;

%F Formula

The row polynomials satisfy (n^3-3*n^2+4*n-2+(n-1)*x-(n-1)*x^2)*P(n,x) = (n-1+(n^3-n^2+n-1)*x-(n-1)*x^3)*P(n-1,x) - (n^2-n+1-x-(n^2-n+2)*x^2+x^3+x^4)*d/dx P(n-1,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[Together[((n - 1 + (n^3 - n^2 + n - 1)*x - (n - 1)*x^3)*P[n - 1, x] - (n^2 - n + 1 - x - (n^2 - n + 2)*x^2 + x^3 + x^4)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 4*n - 2 + (n - 1)*x - (n - 1)*x^2)]];
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] = ((n - 1 + (n^3 - n^2 + n - 1)*x - (n - 1)*x^3)*v[i - 1] - (n^2 - n + 1 - x - (n^2 - n + 2)*x^2 + x^3 + x^4)*deriv(v[i - 1], x))/(n^3 - 3*n^2 + 4*n - 2 + (n - 1)*x - (n - 1)*x^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.

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-525fe5a9d9abc075
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-525fe5a9d9abc075

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

preserver_image derivative_polynomial, shift 0 e-525fe5a9d9abc075

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): chebyshev_t
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check three_term_recurrence e-efd5e57f56b3429d root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-5437345511af69a3 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023