Proof step e-bb5cb95fed86aa9c

preserver_image mobius, shift 0 → concludes about basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(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

• basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x). (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "lower": { "closed": true, "value": "-1" }, "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 2d0b7a56a1b078bed13435e53e290ce24f0dadae5d2d38798fb0700d5a4f7a08, 9,058 bytes (2,445 compressed); download JSON

Cited by

• n-cf1765de6ac2bd4b (one_signed_coefficients)
• n-cf1765de6ac2bd4b (hadamard_kernel)
• n-cf1765de6ac2bd4b (real_rooted)