Proof step e-dee2eaa32f0764ef

preserver_image hadamard_multiplier, shift 0 → concludes about Interpolating the factorial and the powers of 2. Triangle read by rows, T(n, k) for 0 <= k <= n. JSON

Theorem

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

Lean (RealRooted): RealRooted.isMultiplierSequence_inv_factorial

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"hadamard_multiplier\",\"factor\":{\"terms\":[{\"kind\":\"inverse_factorial\"}]}}"
}

Premises

• Triangle of coefficients in expansion of (1+2*x)^n. (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" } } }
• { "claim": "interlacing", "from_index": 1, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 975d63d962b2ecfb2f6ecd281bb351919562b81890c7df46f83e525337eb0a6d, 11,940 bytes (3,064 compressed); download JSON

Cited by

• n-8b66c9f49e6d2f0a (one_signed_coefficients)
• n-8b66c9f49e6d2f0a (hadamard_kernel)
• n-8b66c9f49e6d2f0a (interlacing)
• n-8b66c9f49e6d2f0a (real_rooted)

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.