Proof step

Laguerre’s theorem on differential operators

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

Concludes

shift_x_plus_one of basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).

Premises

one_plus_derivative of Triangle read by rows: A008277 * A007318.
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-7fccaa8f4c74fc62 · 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": 0,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 81a6328c84237c35d8e022a4e8f2f4e58ee8d30539c5d46d3a2103000767f1b2, 10,658 bytes (2,959 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-ab223477daf211c6 (one_signed_coefficients)
n-ab223477daf211c6 (hadamard_kernel)
n-ab223477daf211c6 (real_rooted)

References

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