Proof step `e-71270a30b4492505`

preserver_image derivative_polynomial, shift 0 → concludes about one_plus_derivative of Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= 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\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"
}

Premises

Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n. (sequence)
uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Claims

{ "claim": "real_rooted", "from_index": 1, "region": { "upper": { "closed": true, "value": "0" } } }
{ "claim": "interlacing", "from_index": 2, "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 eb9ee16872ab64fca3d122068861bd235ab7907bea1efc34081d1f6f04c9c39b, 14,502 bytes (3,367 compressed); download JSON

Cited by

n-e6c0c61623a4c862 (real_rooted)
n-e6c0c61623a4c862 (one_signed_coefficients)
n-e6c0c61623a4c862 (hadamard_kernel)
n-e6c0c61623a4c862 (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-71270a30b4492505