Proof step

Laguerre’s theorem on differential operators

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

Concludes

derivative of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).

Premises

Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
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-7907c34d4df91ff5 · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"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 a21a7508f1814889655e45a3f03602cb844b6562a135ccc02a72bbd1430ba057, 13,299 bytes (3,138 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-8f030c9cf6466fb5 (real_rooted)
n-8f030c9cf6466fb5 (one_signed_coefficients)
n-8f030c9cf6466fb5 (hadamard_kernel)
n-8f030c9cf6466fb5 (interlacing)

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.