Proof step
Veronese sections of Pólya frequency sequences
Veronese coefficient section r=2, j=0, shift 0
- Concludes
- veronese_section_2_0 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
- Premises
- Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
- Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
- Theorem
- [ASW52; Edr53; AW24] polytool ?
- Checked
- polytool replayed · Lean not attempted
Technical details
- Step id
e-37358459ee0a1b93· 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-256a20915fe64a0c7f26452cff915a0a11e175f312f9b106016ed77912216f9ee7a, 9,105 bytes (2,444 compressed); downloadpolytool replayed the certificate when the step was written.- Cited by
n-24a064d270958fff(real_rooted)n-24a064d270958fff(one_signed_coefficients)n-24a064d270958fff(hadamard_kernel)n-24a064d270958fff(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