On the properties of alternating invariant functions

2025/05/16 by Zhu, Haiqing, Su Hu, Min-Soo Kim +2
Mathematics · #11B68 #11M06 #33B10 #33B15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2505.10985

openalex publication_date 2025/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Functions satisfying the functional equation ∑_r=0^n-1 (-1)^r f(x+ry, ny) = f(x,y), for any positive odd integer n, are named the alternating invariant functions. Examples of such functions include Euler polynomials, alternating Hurwitz zeta functions and their associated Gamma functions. In this paper, we systematically investigate the fundamental properties of alternating invariant functions. We prove that the set of such functions is closed under translation, reflection, and differentiation. In addition, we define a convolution operation on alternating invariant functions and derive explicit convolution formulas for Euler polynomials and alternating Hurwitz zeta functions, respectively. Furthermore, using distributional relations, we construct new examples of alternating invariant functions, including suitable combinations of trigonometric, exponential, and logarithmic functions, among others.

Citations

• On gamma functions with respect to the alternating Hurwitz zeta functions
• On the properties of invariant functions
• On the Stieltjes constants and gamma functions with respect to alternating Hurwitz zeta functions
• Asymptotic expansions for the alternating Hurwitz zeta function and its derivatives
• Special values and integral representations for the Hurwitz-type Euler zeta functions
• On p-adic Hurwitz-type Euler zeta functions
• Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55)

Related

• Note on multiple q-zeta functions
• Polynomial formulations of Multivariable Arithmetic Progressions of Alternating Powers
• On gamma functions with respect to the alternating Hurwitz zeta functions
• On the Stieltjes constants and gamma functions with respect to alternating Hurwitz zeta functions
• Alternating multiple zeta values, and explicit formulas of some Euler-Apery-type series