Proof step e-1c5b9a0a4dd25652

veronese_section r=2, j=0, shift 0 → concludes about veronese_section_2_0 of Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows. 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) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows. (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 75d31877a0ef3fd208452440b427b15539b6697b036fcf9f64978611286c1ff2, 8,113 bytes (2,291 compressed); download JSON

Cited by

• n-21c6f4e12b73105f (real_rooted)
• n-21c6f4e12b73105f (one_signed_coefficients)
• n-21c6f4e12b73105f (hadamard_kernel)
• n-21c6f4e12b73105f (interlacing)

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