Proof step `e-b8ffc6ab03d37d48`

preserver_image derivative_polynomial, shift 0 → concludes about one_plus_derivative of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)). 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

Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)). (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 e5f13b3b7d4aea7482ea0259d192f595370f9cc6fd18d8c6c687e7a4b39fd88a, 24,224 bytes (5,526 compressed); download JSON

Cited by

n-2ffb0ff2c4cf68d9 (real_rooted)
n-2ffb0ff2c4cf68d9 (one_signed_coefficients)
n-2ffb0ff2c4cf68d9 (hadamard_kernel)
n-2ffb0ff2c4cf68d9 (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-b8ffc6ab03d37d48