Proof step

Laguerre’s theorem on differential operators

Image under a real-rootedness preserver derivative polynomial, shift 0

Concludes

one_plus_derivative of basis_map_A013610 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.

Premises

basis_map_A013610 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Theorem

[SP14] polytool ?

Lean

RealRooted RealRooted.splits_add_C_mul_derivative_all, RealRooted.splits_C_mul_add_derivative, Polynomial.Splits.derivative

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-f1c684d2b7632ae6 · JSON

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\"]}]}"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 0,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

{
  "claim": "interlacing",
  "from_index": 1,
  "kind": "weak",
  "orientation": "unordered"
}

Certificate

preserver_image, SHA-256 37bd2c8393141a4b971c0fd7788b429b7b8a3a9aa11c981202ec64c2259ce21d, 16,639 bytes (3,896 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-2f1b46ca121c31c9 (one_signed_coefficients)
n-2f1b46ca121c31c9 (hadamard_kernel)
n-2f1b46ca121c31c9 (interlacing)
n-2f1b46ca121c31c9 (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.