Proof step
Veronese sections of Pólya frequency sequences
Veronese coefficient section r=2, j=1, shift 0
- Concludes
- veronese_section_2_0 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
- Premises
- veronese parent of Table read by rows: 1 if x = y, 0 otherwise, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
- veronese parent of Table read by rows: 1 if x = y, 0 otherwise, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...
- Theorem
- [ASW52; Edr53; AW24] polytool ?
- Checked
- polytool replayed · Lean not attempted
Technical details
- Step id
e-fdd0396a461b754c· JSON- Operation
- Veronese coefficient section
{ "index_shift": 0, "modulus": 2, "operation": "veronese_section", "residue": 1 } - Claims
{ "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "0" } } }- Certificate
veronese_section, SHA-2564b62607a4d6db6c51a82e8c7da57eac62be12eacade17be3d07511c9fd7ee187, 4,225 bytes (1,241 compressed); downloadpolytool replayed the certificate when the step was written.- Cited by
n-e93cdce9b4ab8c8d(real_rooted)n-e93cdce9b4ab8c8d(one_signed_coefficients)n-e93cdce9b4ab8c8d(hadamard_kernel)
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