Proof step e-f3090409c727b3ad

theorem_check liu_wang → concludes about Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A154537… JSON

Theorem

Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Operation

Theorem check of the node's own rule

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

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": "weak", "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 423e474ef1d9e0d22f15abf5bf6619f17b3f687d19148e8e46db893637c06ccf, 3,322 bytes (1,151 compressed); download JSON

Cited by

• n-945ed98594b7abe2 (real_rooted)
• n-945ed98594b7abe2 (one_signed_coefficients)
• n-945ed98594b7abe2 (hadamard_kernel)
• n-945ed98594b7abe2 (interlacing)

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003