Proof step

Laguerre’s theorem on differential operators

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

Concludes

derivative of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).

Premises

Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).
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-9dfd7cc7abbbb3da · 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": 1,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

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

Certificate

preserver_image, SHA-256 7e5f5708bf4300604614a672ac9a2068c3e8aa699f5299bef0d6f7f6447a9a3b, 13,387 bytes (3,195 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-e47a609ea6f1d1be (real_rooted)
n-e47a609ea6f1d1be (one_signed_coefficients)
n-e47a609ea6f1d1be (hadamard_kernel)
n-e47a609ea6f1d1be (interlacing)

References

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