Higher-Order Cyclotomic Congruences for $q$-Secant and Generalized $q$-Euler Numbers

Abstract

Let $\A(2n)$ denote the set of up--down alternating permutations of $\{1,2,\ldots,2n\}$, and let \[ E_{2n}(q)=\sum_{σ\in\A(2n)}q^{\operatorname{inv}(σ)}. \] Andrews and Foata proved that $E_{2n}(q)\equiv q^{2n(n-1)}\pmod{(1+q)^2}$, and Liu recently obtained the cubic refinement \[ E_{2n}(q)\equiv q^{2n(n-1)}-\binom n2(1+q)^2 \pmod{(1+q)^3}. \] Using the reciprocal generating function for the $q$-secant numbers, a third-order expansion of Gaussian coefficients at $q=-1$, finite differences, and Newton interpolation, we prove the fourth-order refinement \[ E_{2n}(q)\equiv q^{2n(n-1)}-\binom n2(1+q)^2 +\binom n2(2n^2-2n-3)(1+q)^3 \pmod{(1+q)^4}. \] More generally, the recurrence yields an effective procedure for computing the expansion modulo $(1+q)^K$ for any prescribed $K$. We then apply the same local-expansion strategy to the generalized $q$-Euler numbers $E_{pn\mid p}(q)$ of Sagan and Zhang. For every prime $p$, we prove uniform congruences modulo $[p]_q^3$ and $[p]_q^4$; the fourth-order term is governed by a central $q$-Wolstenholme-type quotient associated with ${2p\brack p}_q$. Thus the fourth-order secant congruence is the first case of a general higher-cyclotomic method.