Proof step e-1491609a6fc5f8c9

preserver_image hadamard_multiplier, shift -1 → concludes about Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n. JSON

Theorem

Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

Operation

Image under a real-rootedness preserver

{
  "index_shift": -1,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"hadamard_multiplier\",\"factor\":{\"terms\":[{\"kind\":\"binomial\",\"index_offset\":0,\"coefficient_offset\":1}]}}"
}

Premises

• Even row Pascal-square read by antidiagonals. (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": 1, "region": { "upper": { "closed": true, "value": "0" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 5eb38042fea6a9f935fe5f1f0981348e40df535d1e0605fd0dd9e74e2e27d372, 20,097 bytes (4,003 compressed); download JSON

Cited by

• n-5c0c5bce3aee8449 (real_rooted)
• n-5c0c5bce3aee8449 (one_signed_coefficients)
• n-5c0c5bce3aee8449 (hadamard_kernel)

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.