Proof step e-8de75f6dc51d73d9

theorem_check three_term_recurrence → concludes about Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order). JSON

Theorem

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Operation

Theorem check of the node's own rule

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

Premises

None: this step is a direct theorem check.

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "lower": { "closed": false, "value": "0" }, "upper": { "closed": false, "value": "4" } } }
• { "claim": "interlacing", "from_index": 2, "kind": "strict", "orientation": "previous_interlaces_next" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: theorem_check, SHA-256 4051ea025590773e6d7086c9bd6b4e1ff733999f5f92f9c543279fa79f650f34, 4,293 bytes (1,258 compressed); download JSON

Cited by

• n-f7e7f08472ec987a (interlacing)
• n-f7e7f08472ec987a (real_rooted)
• n-f7e7f08472ec987a (one_signed_coefficients)

References

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