Proof step

Three-term recurrence criterion

Theorem check of the sequence's own rule three term recurrence

Concludes: Riordan array ((1-x)/(1+x), x/(1+x)^2).
Premises: none: a direct theorem check
Theorem: [Sze39, §3.3; Fav35] polytool ?
Checked: polytool replayed · Lean not attempted

Technical details

Step id

e-6a535a5524b82f20 · JSON

Operation

Theorem check of the sequence's own rule

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

Claims

{
  "claim": "real_rooted",
  "from_index": 1,
  "region": {
    "lower": {
      "closed": false,
      "value": "0"
    },
    "upper": {
      "closed": false,
      "value": "4"
    }
  }
}

{
  "claim": "interlacing",
  "from_index": 3,
  "kind": "strict",
  "orientation": "previous_interlaces_next"
}

Certificate

theorem_check, SHA-256 b179f0c23e745425c207555897f9881c7071d56dd33eff23ec403bea457e8819, 5,261 bytes (1,430 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-afb1686ffa6592c0 (interlacing)
n-afb1686ffa6592c0 (real_rooted)
n-afb1686ffa6592c0 (one_signed_coefficients)

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023