Power-partible reduction and congruences for Schröder polynomials

• Chen-Bo Jia ^[1] ; Rong-Hua Wang ^[1] ; Michael X. X. Zhong ^[1]
  1. [1] Tiangong University, China
• Localización: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas ( RACSAM ), ISSN-e 1578-7303, Vol. 118, Nº. 4, 2024
• Idioma: inglés
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• Resumen

  • In this paper, we apply the power-partible reduction to show the following arithmetic properties of large Schröder polynomials Sn(z) and little Schröder polynomials sn(z): for any odd prime p, nonnegative integer r ∈ N, ε ∈ {−1, 1} and z ∈ Z with gcd(p,z(z + 1)) = 1, we have p−1 k=0 (2k + 1) 2r+1εk Sk (z) ≡ 1 (mod p) and p−1 k=0 (2k + 1) 2r+1εk sk (z) ≡ 0 (mod p)