Proof step

Laguerre’s theorem on multiplier sequences

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

Concludes

theta_plus_one of Triangular array: T(n,k) = sqrt(C(n-1,k-1)*C(n-1,k)*C(n,k+1)* C(n+1,k+1)*C(n+1,k)*C(n,k-1)), where C(n,k) = binomial(n,k).

Premises

Triangular array: T(n,k) = sqrt(C(n-1,k-1)*C(n-1,k)*C(n,k+1)* C(n+1,k+1)*C(n+1,k)*C(n,k-1)), where C(n,k) = binomial(n,k).
sequence: Every row P_n, n >= 1, 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-573f0e01daf1be26 · 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": 1,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 63e1ca1a4375b20422e9163fb13a6e7c60f51f3eb713dfaa9b5d50d1616a7e4a, 25,144 bytes (5,867 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-f1b29403ec598a4b (real_rooted)
n-f1b29403ec598a4b (one_signed_coefficients)
n-f1b29403ec598a4b (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.