Proof step e-c585175918782f17

basis_map A159834 → concludes about Coefficient array of Hermite_H(n, (x-1)/sqrt(2))/(sqrt(2))^n. JSON

Theorem

Hermite-type basis map [MW08, Thm 2] polytool ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A159834",
  "operation": "basis_map"
}

Premises

• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward 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": {} }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map, SHA-256 3b0861bef3e0391de8f68ea6e679a108436bf55baae81f0ec7cde2ac18014d8d, 6,410 bytes (1,337 compressed); download JSON

Cited by

• n-f0c4b6406e0da4a3 (real_rooted)

References

1. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741