shifted_legendre

Proved for all n: all roots real, in (0, 1) · interlacing · alternating coefficient signs what this means

Recurrence

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

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

\(P_{2}(x) = 1 - 6\,x + 6\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	-1	2															1
2	1	-6	6														1
3	-1	12	-30	20													1
4	1	-20	90	-140	70												1
5	-1	30	-210	560	-630	252											1
6	1	-42	420	-1680	3150	-2772	924										1
7	-1	56	-756	4200	-11550	16632	-12012	3432									1
8	1	-72	1260	-9240	34650	-72072	84084	-51480	12870								1
9	-1	90	-1980	18480	-90090	252252	-420420	411840	-218790	48620							1
10	1	-110	2970	-34320	210210	-756756	1681680	-2333760	1969110	-923780	184756						1
11	-1	132	-4290	60060	-450450	2018016	-5717712	10501920	-12471030	9237800	-3879876	705432					1
12	1	-156	6006	-100100	900900	-4900896	17153136	-39907296	62355150	-64664600	42678636	-16224936	2704156				1
13	-1	182	-8190	160160	-1701700	11027016	-46558512	133024320	-261891630	355655300	-327202876	194699232	-67603900	10400600			1
14	1	-210	10920	-247520	3063060	-23279256	116396280	-399072960	960269310	-1636014380	1963217256	-1622493600	878850700	-280816200	40116600		1
15	-1	240	-14280	371280	-5290740	46558512	-271591320	1097450640	-3155170590	6544057520	-9816086280	10546208400	-7909656300	3931426800	-1163381400	155117520	1

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Technical details

Node id: n-b835ff4ca4ba76ba · revision 1 · JSON
polytool family: shifted_legendre
Definition source: named_family:shifted_legendre
Fingerprint: f6-d17e9e22f050fa67

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 ?

No OEIS matches.

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: -1,    2;
   2:  1,   -6,     6;
   3: -1,   12,   -30,     20;
   4:  1,  -20,    90,   -140,     70;
   5: -1,   30,  -210,    560,   -630,     252;
   6:  1,  -42,   420,  -1680,   3150,   -2772,     924;
   7: -1,   56,  -756,   4200, -11550,   16632,  -12012,     3432;
   8:  1,  -72,  1260,  -9240,  34650,  -72072,   84084,   -51480,   12870;
   9: -1,   90, -1980,  18480, -90090,  252252, -420420,   411840, -218790,   48620;
  10:  1, -110,  2970, -34320, 210210, -756756, 1681680, -2333760, 1969110, -923780, 184756;

%F Formula

The row polynomials satisfy n*P(n,x) = -(2*n-1-(4*n-2)*x)*P(n-1,x) - (n-1)*P(n-2,x) for n >= 3, with P(1,x) = -1 + 2*x, P(2,x) = 1 - 6*x + 6*x^2.

%t Mathematica

P[1, x_] := -1 + 2*x;
P[2, x_] := 1 - 6*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 1 - (4*n - 2)*x)*P[n - 1, x] - (n - 1)*P[n - 2, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = -1 + 2*x; v[2] = 1 - 6*x + 6*x^2; for(n = 3, N, my(i = n); v[i] = (-(2*n - 1 - (4*n - 2)*x)*v[i - 1] - (n - 1)*v[i - 2])/(n)); 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-b8dd03e52ce06706
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-8b7097919aa29e45, e-b8dd03e52ce06706
Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (0, 1)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-8b7097919aa29e45
Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n(-x) have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-8b7097919aa29e45

Proof chains

2 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

theorem_check three_term_recurrence e-8b7097919aa29e45 root: theorem check

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
theorem_check three_term_recurrence e-b8dd03e52ce06706 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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023