Proof step `e-2f63263d248c26b4`

preserver_image derivative_polynomial, shift 0 → concludes about derivative of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1). 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\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"
}

Premises

Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1). (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": 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 fe721bd6a4498acff3b65be3697d43c1e79e6344490b0f5c0790ea13bc981368, 19,913 bytes (4,401 compressed); download JSON

Cited by

n-4ea7eb31fd1a1554 (real_rooted)
n-4ea7eb31fd1a1554 (one_signed_coefficients)
n-4ea7eb31fd1a1554 (hadamard_kernel)
n-4ea7eb31fd1a1554 (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-2f63263d248c26b4