A053117 → D

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

Recurrence

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

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

\(P_{2}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	2							2
3	0	8						8
4	-4	0	24					20
5	0	-24	0	64				40
6	6	0	-96	0	160			70
7	0	48	0	-320	0	384		112
8	-8	0	240	0	-960	0	896	168

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-776f7a479f11ea52 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-865456471a330200

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 -2)
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 ?

A007290: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:  0;
   2:  2;
   3:  0,   8;
   4: -4,   0,   24;
   5:  0, -24,    0,   64;
   6:  6,   0,  -96,    0,  160;
   7:  0,  48,    0, -320,    0,   384;
   8: -8,   0,  240,    0, -960,     0,   896;
   9:  0, -80,    0,  960,    0, -2688,     0, 2048;
  10: 10,   0, -480,    0, 3360,     0, -7168,    0, 4608;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n+1)*x*P(n-1,x) - (1-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 1)*x*P[n - 1, x] - (1 - x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = ((n + 1)*x*v[i - 1] - (1 - x^2)*deriv(v[i - 1], x))/(n - 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.

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-aa85a20796e296f5
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-aa85a20796e296f5

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-aa85a20796e296f5

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check three_term_recurrence e-b99a39d00579baec root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-dc563681f97c05c6 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

Same fingerprint (1)

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

chebyshev_u → D

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