Proof step `e-05dbfee114f87ac9`

preserver_image euler_polynomial, shift 0 → concludes about theta_plus_one of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k). 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 read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k). (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 5163cc85a68853b5dd94eea9e453de207413186ef209664b46926b25174e809c, 16,366 bytes (3,728 compressed); download JSON

Cited by

n-9e1f147e35ceaa1b (real_rooted)
n-9e1f147e35ceaa1b (one_signed_coefficients)
n-9e1f147e35ceaa1b (hadamard_kernel)
n-9e1f147e35ceaa1b (interlacing)

References

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

Proof step e-05dbfee114f87ac9