Proof step e-95f26075ae445d0d

preserver_image mobius, shift 0 → concludes about Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x). JSON

Theorem

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): route fTransform RealRooted.splits_fPolynomial

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"
}

Premises

• Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial. (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (0, 1)

Claims

• { "claim": "real_rooted", "from_index": 1, "region": {} }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 78e1f8a0df4679185d65239fa39058d0ab7d912b05f068f6fe2c3dc82dac4402, 16,714 bytes (2,880 compressed); download JSON

Cited by

• n-fcc65ac266ed7c2c (real_rooted)