Proof step
Veronese sections of Pólya frequency sequences
Veronese coefficient section r=2, j=0, shift 0
- Concludes
- veronese_section_2_0 of Triangle read by rows: T(n,k) = binomial(k,n-k).
- Premises
- Triangle read by rows: T(n,k) = binomial(k,n-k).
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
- Triangle read by rows: T(n,k) = binomial(k,n-k).
- Theorem
- [ASW52; Edr53; AW24] polytool ?
- Checked
- polytool replayed · Lean not attempted
Technical details
- Step id
e-f1b8ea4179e62f14· JSON- Operation
- Veronese coefficient section
{ "index_shift": 0, "modulus": 2, "operation": "veronese_section", "residue": 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" }- Certificate
veronese_section, SHA-256e70d652cdb7c43bac70021001d70181b99ac58d500cacc3ec97e0efbdf8366bc, 9,063 bytes (2,438 compressed); downloadpolytool replayed the certificate when the step was written.- Cited by
n-579c11d996f64808(real_rooted)n-579c11d996f64808(one_signed_coefficients)n-579c11d996f64808(hadamard_kernel)n-579c11d996f64808(interlacing)
References
- [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
- [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
- [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