Proof step e-81a0a2015f78034f

veronese_section r=2, j=0, shift 0 → concludes about veronese_section_2_0 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2. JSON

Theorem

Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?

Operation

Veronese coefficient section

{
  "index_shift": 0,
  "modulus": 2,
  "operation": "veronese_section",
  "residue": 0
}

Premises

• Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2. (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "0" } } }
• { "claim": "interlacing", "from_index": 1, "kind": "weak", "orientation": "previous_interlaces_next" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: veronese_section, SHA-256 a83f11795ea43f2ebaef1d7125387f3986fcbb9be4a1b94a2a7210ca1d091b60, 8,706 bytes (2,321 compressed); download JSON

Cited by

• n-4b5ab05dec48bd47 (interlacing)
• n-4b5ab05dec48bd47 (real_rooted)
• n-4b5ab05dec48bd47 (one_signed_coefficients)
• n-4b5ab05dec48bd47 (hadamard_kernel)

References

1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
3. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3