A110162 → D

Proved for all n all roots real interlacing (weak)

Recurrence

\(\left(8 - 8\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-2 + x + 4\,n - 2\,n x - 2\,n^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(4\,x - x^{2} - 4\,n x + n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	-4	2													-2
4	9	-12	3												0
5	-16	40	-24	4											4
6	25	-100	105	-40	5										-5
7	-36	210	-336	216	-60	6									0
8	49	-392	882	-840	385	-84	7								7
9	-64	672	-2016	2640	-1760	624	-112	8							-8
10	81	-1080	4158	-7128	6435	-3276	945	-144	9						0
11	-100	1650	-7920	17160	-20020	13650	-5600	1360	-180	10					10
12	121	-2420	14157	-37752	55055	-48048	26180	-8976	1881	-220	11				-11
13	-144	3432	-24024	77220	-137280	148512	-102816	46512	-13680	2520	-264	12			0
14	169	-4732	39039	-148720	316030	-413712	352716	-201552	77805	-20020	3289	-312	13		13
15	-196	6370	-61152	272272	-680680	1058148	-1085280	759696	-368676	123970	-28336	4200	-364	14	-14

Technical details

Node id: n-bfc9def181bdaf52 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-63d46ebead6c4d0e

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   0;
   2:   1;
   3:  -4,     2;
   4:   9,   -12,     3;
   5: -16,    40,   -24,     4;
   6:  25,  -100,   105,   -40,     5;
   7: -36,   210,  -336,   216,   -60,     6;
   8:  49,  -392,   882,  -840,   385,   -84,    7;
   9: -64,   672, -2016,  2640, -1760,   624, -112,    8;
  10:  81, -1080,  4158, -7128,  6435, -3276,  945, -144, 9;

%F Formula

The row polynomials satisfy (2*n^2-8*n+8)*P(n,x) = -(2*n^2-4*n+2-(n^2-2*n+1)*x)*P(n-1,x) - ((4*n-4)*x-(n-1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 4*n + 2 - (n^2 - 2*n + 1)*x)*P[n - 1, x] - ((4*n - 4)*x - (n - 1)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 8*n + 8)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 4*n + 2 - (n^2 - 2*n + 1)*x)*v[i - 1] - ((4*n - 4)*x - (n - 1)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 8*n + 8)); 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, +∞)
by Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 3
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Riordan array ((1-x)/(1+x), x/(1+x)^2).?
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Related ?

Same fingerprint ?

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[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