Proof step e-01e2cd957653d3da

gamma_polynomial scale 1, shift 0 → concludes about Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals. 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 of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals. (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": { "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 25782cdb902ac20eeeddad9c4d6d32feaf98867304ed56721e09fbe731948e4f, 10,538 bytes (2,424 compressed); download JSON

Cited by

• n-c8b3f39d03de798f (real_rooted)
• n-c8b3f39d03de798f (one_signed_coefficients)
• n-c8b3f39d03de798f (hadamard_kernel)