Proof step `e-177701ed6604099d`

preserver_image derivative_polynomial, shift 0 → concludes about Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= 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\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"-1/2\",\"1/2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"-1/2\"],\"denominator\":[\"1\"]}]}"
}

Premises

binomial (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": {} }

Verification and certificate

polytool: replayed (the certificate replayed when the step was written)
Lean: not attempted
certificate: preserver_image, SHA-256 4874d6d906f00336ec12294f3d53747968451fd526809693da9daaa19ec1d45f, 10,843 bytes (2,690 compressed); download JSON

Cited by

n-ccd988a5d83adc8b (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-177701ed6604099d