Proof step e-84c88fc8b81d60d6

theorem_check three_term_recurrence → concludes about Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|. 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 6024053966bbc062c801814b4f81141a2cb1902ac9f67e94cc66676dea3c2378, 3,553 bytes (1,231 compressed); download JSON

Cited by

• n-6ace63f6e12c5046 (interlacing)
• n-6ace63f6e12c5046 (real_rooted)
• n-6ace63f6e12c5046 (one_signed_coefficients)
• n-6ace63f6e12c5046 (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