Proof step e-d52b551add083e42

basis_map A007318 → concludes about basis_map_A007318 of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)). JSON

Theorem

Geometric basis map (elementary) polytool ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A007318",
  "operation": "basis_map"
}

Premises

• Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)). (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": 1, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map, SHA-256 8bd24a2fe4c679bb0a60e949fce76ea3e24548a9d845dce138d7dd72f9e835e9, 13,622 bytes (2,595 compressed); download JSON

Cited by

• n-0135f5d413e6cc58 (real_rooted)
• n-0135f5d413e6cc58 (one_signed_coefficients)
• n-0135f5d413e6cc58 (hadamard_kernel)
• n-0135f5d413e6cc58 (interlacing)