Integral and arithmetic structures of alternating (zigzag) numbers $A_n$

TL;DR

This paper unifies and extends various combinatorial, analytic, and arithmetic representations of alternating (zigzag) numbers, revealing their deep structural connections and deriving new integral formulas, spectral interpretations, and congruence relations.

Contribution

It introduces new integral representations, spectral interpretations, and congruence relations for zigzag numbers, bridging combinatorial, analytic, and arithmetic perspectives.

Findings

Derived contour and Laplace integral representations of $A_n$

Expressed $A_{2n+1}$ as a hyperbolic integral involving spectral moments

Established congruence relations modulo primes for $A_n$

Abstract

The alternating (zigzag) numbers $A_n$, counting the ascending alternating permutations of $\left\{1,\cdots,n\right\}$ and defined by the exponential generating function $\tan x+\sec x$, admit several classical combinatorial and analytic representations. In this work we unify and extend three complementary structures of $A_n$. First, starting from the Stirling number expansion of zigzag numbers, we derive a contour integral representation, as well as a positive Laplace-type integral representation $$ A_n = 2^n \int_0^\infty e^{-y} f_n(y)\, dy, \qquad f_n(y) := \sum_{k=0}^{n} (-1)^k S(n,k) \left(\frac{y}{2}\right)^k, $$ where the kernel $f_n(y)$ is the polynomial generating function of Stirling numbers. A continuous interpolation of the discrete product (falling factorial) is introduced subsequently. This provides a direct analytic bridge between set partitions and Laplace asymptotics. Second, using the partial fraction expansion of $\tan$, we obtain the well-known hyperbolic integral representation $$ A_{2n+1}=\frac{1}{\pi}\int_0^\infty\frac{y^{2n+1}}{\sinh(y/2)}\,dy, $$ equivalently expressed in classical $\cosh$ form for $A_{2n}$. This representation interprets zigzag numbers as spectral moments associated with half-integer poles. The connection with Fourier analysis and Mellin transforms is also outlined. Finally, combining spectral expansions with Stirling identities, we derive congruence relations modulo primes for $A_n$. These results exhibit a dual analytic-combinatorial structure of zigzag numbers, linking partition expansions, trigonometric spectra, and arithmetic properties.