Proof step

Laguerre’s theorem on differential operators

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

Concludes

derivative of basis_map_A007318 of Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Premises

basis_map_A007318 of Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
sequence: Every row P_n, n >= 0, 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-e43a4ecad5cd6484 · JSON

Operation

Image under a real-rootedness preserver

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

Claims

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

Certificate

preserver_image, SHA-256 f51954f2888533e527c6750f2027aa414dee1d78492abf5156ee5f070c529a8c, 12,265 bytes (3,446 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-9051098c52bc0842 (real_rooted)
n-9051098c52bc0842 (one_signed_coefficients)
n-9051098c52bc0842 (hadamard_kernel)

References

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