Proof step e-b7b551b1755ad9dc

basis_map_theorem → concludes about Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) =… JSON

Theorem

Index-dependent basis map [Bre89, Thm 2.4.3; MW08, Thm 2] polytool RealRooted basisMap_factorization_of_scalarIndex_recurrence RealRooted rayBasisMap_maps_realRootedIn_iff ?

Operation

Basis map theorem for the node's own rule

{
  "operation": "basis_map_theorem"
}

Premises

None: this step is a direct theorem check.

Claims

• { "claim": "basis_map_preserver", "input_interval": "(-inf, 0]", "output_interval": "(-inf, 0]", "statement": "(-inf, 0] -> (-inf, 0] through R_s, s = 1, and U_g" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map_theorem, SHA-256 08984e77a22c75a58504bbb3ac6482ef21fb2c4e145462151499dc138338eac9, 2,673 bytes (1,102 compressed); download JSON

Cited by

• n-914700b1060feeb2 (basis_map_preserver)

References

1. [Bre89] Francesco Brenti. Unimodal, log-concave and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc., 81(413):viii+106, 1989. doi:10.1090/memo/0413
2. [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