A103327 → D → D

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Triangle begins:
  0:     0;
  1:     0;
  2:     2;
  3:    42,      6;
  4:   252,    216,      12;
  5:   924,   1980,     660,      20;
  6:  2574,  10296,    8580,    1560,     30;
  7:  6006,  38610,   60060,   27300,   3150,     42;
  8: 12376, 116688,  291720,  247520,  71400,   5712,   56;
  9: 23256, 302328, 1108536, 1511640, 813960, 162792, 9576, 72;

The row polynomials satisfy (2*n^2-9*n+10)*P(n,x) = (2*n^2+n+(6*n^2-21*n+12)*x)*P(n-1,x) + ((4*n-4)*x-(4*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + n + (6*n^2 - 21*n + 12)*x)*P[n - 1, x] + ((4*n - 4)*x - (4*n - 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 9*n + 10)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 + n + (6*n^2 - 21*n + 12)*x)*v[i - 1] + ((4*n - 4)*x - (4*n - 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 9*n + 10)); 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)

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 derivative of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Three-term recurrence criterion (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023