Proof step

Laguerre’s theorem on multiplier sequences

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

Concludes

theta_plus_one_squared of Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.

Premises

Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.
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-f07788e7e2379024 · JSON

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\"]}]}"
}

Claims

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

Certificate

preserver_image, SHA-256 07f32c2cbb80aa78b594138921e984d3a29cd435564d02b7c1e3a23f41378691, 10,271 bytes (2,863 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-2a2ed4d5f9d376c7 (real_rooted)
n-2a2ed4d5f9d376c7 (one_signed_coefficients)
n-2a2ed4d5f9d376c7 (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.