Proof step e-907e3c6c6275331d

preserver_image euler_polynomial, shift 0 → concludes about Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2). JSON

Theorem

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

Lean (RealRooted): RealRooted.isMultiplierSequence_eval_of_roots_nonpos, RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos, RealRooted.isMultiplierSequence_iff_preserves

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"
}

Premises

• Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n. (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": { "upper": { "closed": true, "value": "0" } } }
• { "claim": "interlacing", "from_index": 2, "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 4fd18a04600a4fc7d277d5baf0da6a74b510a60684c83662e1e44a008c698bbd, 16,411 bytes (3,350 compressed); download JSON

Cited by

• n-f25b32370a8d627e (real_rooted)
• n-f25b32370a8d627e (one_signed_coefficients)
• n-f25b32370a8d627e (hadamard_kernel)
• n-f25b32370a8d627e (interlacing)

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.