Proof step

Liu–Wang theorem with a lag row

Theorem check of the sequence's own rule liu wang derivative lag

Concludes: Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(…
Premises: none: a direct theorem check
Theorem: [LW07, Thm 2.1] RealRooted ?
Lean: RealRooted route halfGrowth RealRooted.derivLag_strictInterl_of_nonnegCoeffs, RealRooted.derivLag_strictInterl_and_noCommonRoot, RealRooted.derivLag_eval_zero_pos
Checked: polytool replayed · Lean not attempted

Technical details

Step id

e-dff7465450f7ae66 · JSON

Operation

Theorem check of the sequence's own rule

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

Claims

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

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

Certificate

theorem_check, SHA-256 4446923d0c449f736ad808d3660a400f2732068f9b55e43365df5dd8ffa3b238, 6,173 bytes (1,707 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