Proof step

Laguerre’s theorem on differential operators

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

Concludes

derivative of Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.

Premises

Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.
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-3e915b081502f726 · 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 16725ee9d5b29de2731fe9d56a1257e4803c36f545bb02520c32db28d6e8bb27, 18,347 bytes (5,598 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-36292cabe246c808 (real_rooted)
n-36292cabe246c808 (one_signed_coefficients)
n-36292cabe246c808 (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.