Proof step e-7d7c781fa7a59405

gamma_polynomial scale 1, shift 0 → concludes about gamma_polynomial of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!). 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

• Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!). (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 c244550b5f4a888564aa8219f4e01e416a3fdc41bcd56362e354edda581d69f1, 9,851 bytes (2,268 compressed); download JSON

Cited by

• n-de8575ce50ceeb5a (real_rooted)
• n-de8575ce50ceeb5a (one_signed_coefficients)
• n-de8575ce50ceeb5a (hadamard_kernel)