Proof step

Liu–Wang theorem for three-term recurrences

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

Concludes: Regular triangle read by rows. T(n, k) = {{n, k}}, where {{n, k}} are the second order Stirling set numbers (or second order Stirling numbers). T(n, k) for 0 <…
Premises: none: a direct theorem check
Theorem: [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ?
Checked: polytool replayed · Lean not attempted

Technical details

Step id

e-fe357ce1dbcb978c · JSON

Operation

Theorem check of the sequence's own rule

{
  "operation": "theorem_check",
  "theorem": "liu_wang_three_term"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 3,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

{
  "claim": "interlacing",
  "from_index": 4,
  "kind": "weak",
  "orientation": "previous_interlaces_next"
}

Certificate

theorem_check, SHA-256 f26f887bfc3b11574402a394b86b31d8b257ba4fbdf2571f2d677bd08884dc71, 7,223 bytes (1,800 compressed); download

polytool replayed the certificate when the step was written.

Cited by

References

[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