Proof step
Pólya–Schur theorem on multiplier sequences
Hadamard product with a multiplier sequence 1/k!
- Concludes
- Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
- Premises
- hadamard_factor parent of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
- hadamard_factor parent of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
- Theorem
- [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ?
- Checked
- polytool replayed · Lean not attempted
Technical details
- Step id
e-25cc044b20673aa9· JSON- Operation
- Hadamard product with a multiplier sequence
{ "factor": "1/k!", "operation": "hadamard_factor" } - Claims
{ "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "0" } } }- Certificate
hadamard_factor, SHA-2563a82bb0932083cce5442daa2c6e1334d45c0453063e428b032c97278103e28d8, 10,662 bytes (2,345 compressed); downloadpolytool replayed the certificate when the step was written.- Cited by
n-511a2225fa2191c9(real_rooted)n-511a2225fa2191c9(one_signed_coefficients)n-511a2225fa2191c9(hadamard_kernel)
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.