Proof step e-781e4104e926c42c

basis_map A008277 → concludes about basis_map_A008277 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)). JSON

Theorem

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

Operation

Image under a basis map u^k -> P_k

{
  "map": "A008277",
  "operation": "basis_map"
}

Premises

• Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)). (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: basis_map, SHA-256 f76d729ad44a73dacfffc4a37a54283beecaf05a7c48089a85021cda284b3e31, 12,843 bytes (2,339 compressed); download JSON

Cited by

• n-07e5e6cd238b2e3e (real_rooted)
• n-07e5e6cd238b2e3e (one_signed_coefficients)
• n-07e5e6cd238b2e3e (hadamard_kernel)

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