Proof step `e-0c9044372fd705ed`

preserver_image derivative_polynomial, shift 0 → concludes about Number triangle T(n,k)=C(n,k)C(2,n-k). 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\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1/2\"],\"denominator\":[\"1\"]}]}"
}

Premises

a(n) = 1 if n is of the form m(m+3)/2, otherwise 0. (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" } } }

Verification and certificate

polytool: replayed (the certificate replayed when the step was written)
Lean: not attempted
certificate: preserver_image, SHA-256 1fd0e8d44a26228023c2e2f218400a6cb694f37ec395d8a86cce7cd040cbb4eb, 7,486 bytes (1,913 compressed); download JSON

Cited by

n-1815d3e1c9c57d5b (real_rooted)
n-1815d3e1c9c57d5b (one_signed_coefficients)
n-1815d3e1c9c57d5b (hadamard_kernel)

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-0c9044372fd705ed