Proof step

Laguerre’s theorem on differential operators

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

Concludes

one_plus_derivative of Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.

Premises

Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.
sequence: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Theorem

[SP14] polytool ?

Lean

RealRooted RealRooted.splits_add_C_mul_derivative_all, RealRooted.splits_C_mul_add_derivative, Polynomial.Splits.derivative

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-a8b72947c1839ff2 · JSON

Operation

Image under a real-rootedness preserver

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

Claims

{
  "claim": "real_rooted",
  "from_index": 1,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

{
  "claim": "interlacing",
  "from_index": 2,
  "kind": "weak",
  "orientation": "unordered"
}

Certificate

preserver_image, SHA-256 bb1752f8a484a1e57fb49a1a52e0663ae0f664e61d5f8cfc3db0b8286e93b3a2, 19,601 bytes (4,264 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-614aff0891e33f9b (real_rooted)
n-614aff0891e33f9b (one_signed_coefficients)
n-614aff0891e33f9b (hadamard_kernel)
n-614aff0891e33f9b (interlacing)

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.