Proof step `e-5d20fd95bc57193f`

preserver_image derivative_polynomial, shift 0 → concludes about one_plus_derivative of Triangle of coefficients in expansion of (1+4x)^n. 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\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"
}

Premises

Triangle of coefficients in expansion of (1+4x)^n. (sequence)
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Claims

{ "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "0" } } }
{ "claim": "interlacing", "from_index": 1, "kind": "weak", "orientation": "unordered" }

Verification and certificate

polytool: replayed (the certificate replayed when the step was written)
Lean: not attempted
certificate: preserver_image, SHA-256 9ce786d910e8ef160dc2bb28b35d7a48b782f40b7feefe1fba515d072893c417, 11,155 bytes (2,904 compressed); download JSON

Cited by

n-bf9d0f008400b3c6 (real_rooted)
n-bf9d0f008400b3c6 (one_signed_coefficients)
n-bf9d0f008400b3c6 (hadamard_kernel)
n-bf9d0f008400b3c6 (interlacing)

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-5d20fd95bc57193f