Proof step `e-6f899a70e4c27d14`

preserver_image derivative_polynomial, shift 0 → concludes about derivative of Triangle of coefficients of cos(x)^n in polynomial for cos(nx). JSON

Theorem

Laguerre’s theorem on differential operators [SP14] polytool ?

Lean (RealRooted): RealRooted.splits_add_C_mul_derivative_all, RealRooted.splits_C_mul_add_derivative, Polynomial.Splits.derivative

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\"]}]}"
}

Premises

Triangle of coefficients of cos(x)^n in polynomial for cos(nx). (sequence)
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)

Claims

{ "claim": "real_rooted", "from_index": 0, "region": {} }

Verification and certificate

polytool: replayed (the certificate replayed when the step was written)
Lean: not attempted
certificate: preserver_image, SHA-256 0591ef23d17a4e0d55cbe3966b3a2b2358c92c581ac711f615dc7b23070eb2e0, 14,211 bytes (2,726 compressed); download JSON

Cited by

n-02ad05e8959bc17f (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.

Proof step e-6f899a70e4c27d14