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

TL;DR

The paper studies the alternating (zigzag) numbers $A_n$ defined by $tan x + sec x$ and unifies three analytic representations: a contour integral from the Stirling-number expansion, a positive Laplace-type integral with a Gamma-based interpolation of falling factorials, and a hyperbolic spectral integral from the partial fraction expansion of $tan$. It also derives arithmetic properties of $A_n$ by exploiting Stirling-number identities to obtain congruences modulo primes and, in some cases, prime powers. The contributions include a Stirling-based contour integral, a Laplace-type integral bridging partitions and continuous kernels, and a hyperbolic integral linked to Fourier/Mellin transforms, together with explicit modular relations. This unified framework links combinatorial partitions, spectral analysis, and arithmetic properties of $A_n$, with implications for asymptotics, transforms, and deeper number-theoretic structure.

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 $$ A_{2n+1}=\frac{1}π\int_0^\infty\frac{y^{2n+1}}{\sinh(y/2)}\,dy, $$ 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.