Proof step `e-d32ce5c030354d3d`

preserver_image derivative_polynomial, shift 0 → concludes about one_plus_derivative of Triangle of coefficients of shifted Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in decreasing order). 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

Triangle of coefficients of shifted Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in decreasing order). (sequence)
uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)

Claims

{ "claim": "real_rooted", "from_index": 1, "region": {} }

Verification and certificate

polytool: replayed (the certificate replayed when the step was written)
Lean: not attempted
certificate: preserver_image, SHA-256 fa5a09b6f83d4ffa109de372df949265b9a70fb2d9dfc2c4bd28143360f3a935, 14,775 bytes (2,930 compressed); download JSON

Cited by

n-0de906cd6e92b6be (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-d32ce5c030354d3d