chebyshev_t → x ↦ x+1 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	4	4						8
3	9	24	12					45
4	16	80	96	32				224
5	25	200	420	320	80			1045
6	36	420	1344	1728	960	192		4680
7	49	784	3528	6720	6160	2688	448	20377

All 15 rows

Technical details

Node id: n-45560eec68fca4d5 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-d6d34c33242db792

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): yes
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  0;
  1:  1;
  2:  4,    4;
  3:  9,   24,    12;
  4: 16,   80,    96,    32;
  5: 25,  200,   420,   320,     80;
  6: 36,  420,  1344,  1728,    960,    192;
  7: 49,  784,  3528,  6720,   6160,   2688,   448;
  8: 64, 1344,  8064, 21120,  28160,  19968,  7168,  1024;
  9: 81, 2160, 16632, 57024, 102960, 104832, 60480, 18432, 2304;

%F Formula

The row polynomials satisfy (n^2-2*n+1)*P(n,x) = (n^2+n^2*x)*P(n-1,x) + (2*n*x+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 + n^2*x)*P[n - 1, x] + (2*n*x + 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 + n^2*x)*v[i - 1] + (2*n*x + 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from shift_x_plus_one of chebyshev_t?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from chebyshev_t?
    
    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.
[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