A082985 → reverse → D

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Submission text ?

Triangle begins:
  0:   0;
  1:   1;
  2:   5,    2;
  3:  14,   14,    3;
  4:  30,   54,   27,     4;
  5:  55,  154,  132,    44,    5;
  6:  91,  364,  468,   260,   65,    6;
  7: 140,  756, 1350,  1100,  450,   90,    7;
  8: 204, 1428, 3366,  3740, 2210,  714,  119,   8;
  9: 285, 2508, 7524, 10868, 8645, 3990, 1064, 152, 9;

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

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 3*n + 1 + (n^2 + n)*x)*P[n - 1, x] + (4*n*x + n*x^2)*D[P[n - 1, x], x])/(2*n^2 - 3*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

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

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 reversal of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
                • ↑ Three-term recurrence criterion (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from shift_x_plus_one of Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q…?
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where…?
                    • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A111125 → D

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