Proof step

Laguerre’s theorem on differential operators

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

Concludes

derivative of Riordan array ((1-x)/(1+x), x/(1+x)^2).

Premises

Riordan array ((1-x)/(1+x), x/(1+x)^2).
sequence: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)

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-b38122b7d5253ee9 · 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": {}
}

{
  "claim": "interlacing",
  "from_index": 3,
  "kind": "weak",
  "orientation": "unordered"
}

Certificate

preserver_image, SHA-256 7d0890de7b959d0f3cd51df4cb8d85ef65088ce43563cb033a02d878152b7b55, 22,754 bytes (3,689 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-bfc9def181bdaf52 (real_rooted)
n-bfc9def181bdaf52 (interlacing)

References

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