Proof step `e-2036104bf9bd843b`

preserver_image derivative_polynomial, shift 0 → concludes about one_plus_derivative of basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(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

basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + … (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 1164b24cc47f7131539bb7f73fe3e8f35dfa5a9635898b197e7a84423bc374e9, 20,305 bytes (4,528 compressed); download JSON

Cited by

n-0506e8505d88de13 (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-2036104bf9bd843b