Proof step e-c0dcd56725ae4481

theorem_check three_term_recurrence → concludes about Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ). 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": 0, "region": { "upper": { "closed": false, "value": "0" } } }
• { "claim": "interlacing", "from_index": 1, "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 731e4750c540a737334ba6256f1a616ac22e6422eee6c3c200675e6564bbd9ba, 4,359 bytes (1,331 compressed); download JSON

Cited by

• n-9aec4822212db1dc (interlacing)
• n-9aec4822212db1dc (real_rooted)
• n-9aec4822212db1dc (one_signed_coefficients)
• n-9aec4822212db1dc (hadamard_kernel)

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