Proof step `e-c9e97b9a66627f15`

preserver_image derivative_polynomial, shift 0 → concludes about derivative of Triangle read by rows: T(n,k), 0 <= k <= n, gives the coefficients of the Charlier polynomials (with parameter a=1), ordered by rising powers. 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), 0 <= k <= n, gives the coefficients of the Charlier polynomials (with parameter a=1), ordered by rising powe… (sequence)
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)

Claims

{ "claim": "real_rooted", "from_index": 0, "region": {} }
{ "claim": "interlacing", "from_index": 1, "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 31b5fec3d481284dfce4c7b824b52b61a319898390722198c3b9e0cfcc4b9604, 13,831 bytes (2,902 compressed); download JSON

Cited by

n-68463e7ee4b23199 (real_rooted)
n-68463e7ee4b23199 (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-c9e97b9a66627f15