Proof step

Laguerre’s theorem on multiplier sequences

Image under a real-rootedness preserver euler polynomial, shift 0

Concludes

theta_plus_one of Irregular triangle read by rows, Stirling numbers of the second kind: columns shifted to allow (1, 1, 2, 2, 3, 3, ...) terms per row.

Premises

Irregular triangle read by rows, Stirling numbers of the second kind: columns shifted to allow (1, 1, 2, 2, 3, 3, ...) terms per row.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Theorem

[SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ?

Lean

RealRooted RealRooted.isMultiplierSequence_eval_of_roots_nonpos, RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos, RealRooted.isMultiplierSequence_iff_preserves

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-5eb8bb2bfa60e69d · JSON

Operation

Image under a real-rootedness preserver

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

Claims

{
  "claim": "real_rooted",
  "from_index": 0,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 08ffddd80a0225e99b36d566d89165176167ac2e4757fa1bdfc2bc8074628109, 10,708 bytes (2,844 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-33143131ebb2fed2 (real_rooted)
n-33143131ebb2fed2 (one_signed_coefficients)
n-33143131ebb2fed2 (hadamard_kernel)

References

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