Real-rooted atlas

Proof step

Liu–Wang theorem for three-term recurrences

Theorem check of the sequence's own rule liu wang three term

Concludes
Riordan array ((1-x)/(1-2*x), x(1-x)/(1-2*x)^2).
Premises
none: a direct theorem check
Theorem
[LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ?
Lean
RealRooted RealRooted.threeTerm_interlaces_of_window
Checked
polytool replayed · Lean not attempted
Technical details
Step id
e-b3ebc341e0279939 · JSON
Operation
Theorem check of the sequence's own rule
{
  "operation": "theorem_check",
  "theorem": "liu_wang_three_term"
}
Claims
{
  "claim": "real_rooted",
  "from_index": 0,
  "region": {
    "lower": {
      "closed": true,
      "value": "-4"
    }
  }
}
{
  "claim": "interlacing",
  "from_index": 2,
  "kind": "weak",
  "orientation": "previous_interlaces_next"
}
Certificate
theorem_check, SHA-256 13e45c161c28a702bbccca9c86ec7b4b9d063e733ef493ab3bb3e471aefe6f2d, 4,405 bytes (1,439 compressed); download
polytool replayed the certificate when the step was written.
Cited by

References

  1. [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