Proof step e-10ee054377433864

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0). JSON

Theorem

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

Operation

Gamma polynomial of a palindromic parent

{
  "index_shift": 0,
  "operation": "gamma_polynomial",
  "scale": "1"
}

Premises

• Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=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: gamma_polynomial, SHA-256 b275c8d6a5c629467360a41ab2bd881f921a7c8351f41b2500d46615f793b220, 15,462 bytes (3,192 compressed); download JSON

Cited by

• n-6cf134ccfa9a5641 (real_rooted)
• n-6cf134ccfa9a5641 (one_signed_coefficients)
• n-6cf134ccfa9a5641 (hadamard_kernel)