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Peng-Cheng Hang

Publications and source records attributed to Peng-Cheng Hang.

8 recordsLinked to original sources
arXiv2026

Fourier transform pairs and Eisenstein-type series related to Jacobi elliptic functions

We compute Fourier transforms of functions expressed as a ratio of one of the Jacobi elliptic functions divided by $\sinh(πx)$ or $\cosh(πx)$. In many cases, the resulting Fourier transform remains within the same class of functions. Applying the Mellin transform, we obtain sixteen Eisenstein-type series $ζ_{j,l}(s,τ)$, for which we establish several results: analytic continuation with respect to the variable $s$, a functional equation connecting $ζ_{j,l}(s,τ)$ and $ζ_{l,j}(1-s,-1/τ)$, and explicit expressions for $ζ_{j,l}(s,τ)$ when $s$ runs through a sequence of positive even or odd integers.

math.CA↗
arXiv2026

Asymptotic expansions of the Humbert Function $Φ_1$ and their applications

This paper systematically studies the asymptotics of Humbert's bivariate confluent hypergeometric function $Φ_1[a,b;c;x, y]$. Specifically, we establish explicit asymptotic expansions in five distinct regimes: (i) $x\to\infty$; (ii) $y\to\infty$; (iii) $x\to\infty,\,y\to\infty$; (iv) $x$ or $y$ small, $xy$ fixed; and (v) $x\to 1$, $y$ fixed. The utility of these expansions is illustrated through concrete applications in the theory of Saran's hypergeometric function $F_M$, the Glauber-Ising model, and the theory of Prabhakar-type fractional integral operators. Several potential directions for future work are also outlined.

math.CA↗
arXiv2025

Complete asymptotic expansions of the Humbert function $Ψ_1$ for two large arguments

In our recent work [SIGMA \textbf{20} (2024), 074, 13 pages], the leading behaviour of the Humbert function $Ψ_1[a,b;c,c';x,y]$ when $x\to\infty$ and $y\to +\infty$ has been derived in a direct and simple manner. In this paper, we obtain the complete asymptotics of $Ψ_1$ in the general case $x,y\to\infty$ along a new path. Indeed, our proof is based on a sharp estimate on ${}_2F_2[a,b-n;c,d-n;z]$, which is valid uniformly for $n\in\mathbb{Z}_{\geqslant 0}$ and large $z$.

math.CA↗
arXiv2025

Asymptotics of the Humbert functions $Ψ_1$ and $Ψ_2$

A compilation of new results on the asymptotic behaviour of the Humbert functions $Ψ_1$ and $Ψ_2$, and also on the Appell function $F_2$, is presented. As a by-product, we confirm a conjectured limit which appeared recently in the study of the $1D$ Glauber-Ising model. We also propose two elementary asymptotic methods and confirm through some illustrative examples that both methods have great potential and can be applied to a large class of problems of asymptotic analysis. Finally, some directions of future research are pointed out in order to suggest ideas for further study.

math.CA↗
arXiv2024

Note on the $a$-points of the Riemann zeta function

For any $a\in\mathbb{C}$, the zeros of $ζ(s)-a$, denoted by $ρ_a=β_a+iγ_a$, are called $a$-points of the Riemann zeta function $ζ(s)$. In this paper, we reformulate some basic results about the $a$-points of $ζ(s)$ shown by Garunkštis and Steuding. We then deduce an asymptotic of the sum \[S_T(a,δ)=\sum_{τ<γ_a\leqslant T}ζ'(ρ_a+iδ)X^{ρ_a},\quad T\to\infty,\] where $0\neδ=\frac{2πα}{\log\frac{T}{2πX}}\ll 1$, and $X>0$ and $τ\geqslant|δ|+1$ are fixed. We also find the interesting varied behavior of $S_T(a,δ)$ in different $X$ ranges, which is more complicated than those described before by Gonek and Pearce-Crump.

math.NT↗
arXiv2024

Asymptotics of the Humbert Function $Ψ_1$ for Two Large Arguments

Recently, Wald and Henkel (2018) derived the leading-order estimate of the Humbert functions $Φ_2$, $Φ_3$ and $Ξ_2$ for two large arguments, but their technique cannot handle the Humbert function $Ψ_1$. In this paper, we establish the leading asymptotic behavior of the Humbert function $Ψ_1$ for two large arguments. Our proof is based on a connection formula of the Gauss hypergeometric function and Nagel's approach (2004). This approach is also applied to deduce asymptotic expansions of the generalized hypergeometric function $_pF_q$ $(p\leqslant q)$ for large parameters, which are not contained in NIST handbook.

math.CA↗
arXiv2024

Asymptotics of Saran's hypergeometric function $F_K$

In this paper, we first establish asymptotic expansions of the Humbert function $Ψ_1$ for one large variable. The resulting expansions are then used to derive an asymptotic expansion of Saran's hypergeometric function $F_K$ when two of its variables become simultaneously large.

math.CA↗