Proof step e-96faac355005fea1

preserver_image euler_polynomial, shift 0 → concludes about theta_plus_one_squared of Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal… JSON

Theorem

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

Lean (RealRooted): RealRooted.isMultiplierSequence_eval_of_roots_nonpos, RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos, RealRooted.isMultiplierSequence_iff_preserves

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"
}

Premises

• Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1… (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" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 2ed9a90cf687116d3a3eb73b12b01aa4c81065c07299ccfd36b979790cc28ea5, 16,948 bytes (4,432 compressed); download JSON

Cited by

• n-ae7239a4b292aee3 (real_rooted)
• n-ae7239a4b292aee3 (one_signed_coefficients)
• n-ae7239a4b292aee3 (hadamard_kernel)

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.