Proof step e-537ef91063de36e5

theorem_check three_term_recurrence → concludes about Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-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": 1, "region": { "upper": { "closed": false, "value": "0" } } }
• { "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 bddb0a93d1ca09ed919eb8d7cd9e2e2ff4300efb4d33d77c8c4368784e047374, 3,459 bytes (1,196 compressed); download JSON

Cited by

• n-976cdb6e4a693ec8 (hadamard_kernel)
• n-976cdb6e4a693ec8 (interlacing)
• n-976cdb6e4a693ec8 (real_rooted)
• n-976cdb6e4a693ec8 (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