Proof step `e-168270346f6f7c93`

preserver_image derivative_polynomial, shift 0 → concludes about Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k). 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\",\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"-1\",\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"-1\"],\"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 c63a86c4341711ecda19064b1a12563f7f94a03b0943a10bec87d83d133ac56e, 10,944 bytes (2,637 compressed); download JSON

Cited by

n-14d5814e0f152cf3 (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-168270346f6f7c93