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 of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= 1 and 0 <= k <= n-1, read by ro…

Premises

Triangular array of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= 1 and 0 <= k <= n-1, read by …
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-25592cbd0b4a7710 · 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 24c61d8e7dd7d6c9bf6cbbff0e6ce3576a1810b1ce3b7982ee60ca32bf0a4d8f, 16,434 bytes (4,344 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-7d7a2499dfefbc27 (one_signed_coefficients)
n-7d7a2499dfefbc27 (hadamard_kernel)
n-7d7a2499dfefbc27 (real_rooted)

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.