Proof step

Laguerre’s theorem on multiplier sequences

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

Concludes

theta_plus_one_squared of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).

Premises

Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
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-7065d7d287e9b5bf · 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"
    }
  }
}

{
  "claim": "interlacing",
  "from_index": 1,
  "kind": "weak",
  "orientation": "unordered"
}

Certificate

preserver_image, SHA-256 fab520cb58cb0d77c6c4b70075a2f76d3e4aadd9bc0f0e59d40bd67aa72bc625, 15,241 bytes (3,593 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-4b3b67524c8f9abc (real_rooted)
n-4b3b67524c8f9abc (one_signed_coefficients)
n-4b3b67524c8f9abc (hadamard_kernel)
n-4b3b67524c8f9abc (interlacing)

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.