A207607 → basis A013610 → D

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Triangle begins:
  0:     0;
  1:     6;
  2:    27,     36;
  3:    99,    270,     162;
  4:   333,   1368,    1701,      648;
  5:  1065,   5832,   11097,     8748,     2430;
  6:  3294,  22518,   57834,    69984,    40095,     8748;
  7:  9948,  81486,  263736,   432540,   380295,   170586,   30618;
  8: 29511, 281628, 1100061,  2282580,  2722815,  1872072,  688905,  104976;
  9: 86337, 940464, 4303773, 10815444, 16347825, 15247764, 8588349, 2676888, 354294;

The row polynomials satisfy (6*n^3-45*n^2+45*n-6+(36*n^3-108*n^2+108*n-36)*x)*P(n,x) = (9*n^3-63*n^2-18*n+(63*n^3-144*n^2-27*n)*x+(54*n^3-54*n^2)*x^2)*P(n-1,x) + (5*n^2-20*n-15+(48*n^2-102*n-54)*x+(117*n^2-144*n-27)*x^2+(54*n^2-54*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.

P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^3 - 63*n^2 - 18*n + (63*n^3 - 144*n^2 - 27*n)*x + (54*n^3 - 54*n^2)*x^2)*P[n - 1, x] + (5*n^2 - 20*n - 15 + (48*n^2 - 102*n - 54)*x + (117*n^2 - 144*n - 27)*x^2 + (54*n^2 - 54*n)*x^3)*D[P[n - 1, x], x])/(6*n^3 - 45*n^2 + 45*n - 6 + (36*n^3 - 108*n^2 + 108*n - 36)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^3 - 63*n^2 - 18*n + (63*n^3 - 144*n^2 - 27*n)*x + (54*n^3 - 54*n^2)*x^2)*v[i - 1] + (5*n^2 - 20*n - 15 + (48*n^2 - 102*n - 54)*x + (117*n^2 - 144*n - 27)*x^2 + (54*n^2 - 54*n)*x^3)*deriv(v[i - 1], x))/(6*n^3 - 45*n^2 + 45*n - 6 + (36*n^3 - 108*n^2 + 108*n - 36)*x)); 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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013610 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.?
        • 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
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
                • ↑ Three-term recurrence criterion (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
                • ↑ Three-term recurrence criterion (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Three-term recurrence criterion (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