论文标题
c-多项式和LC功能：迈向Hurwitz Zeta功能的概括
C-polynomials and LC-functions: towards a generalization of the Hurwitz zeta function
论文作者
论文摘要
令$ f（t）= \ sum_ {n = 0}^{+\ infty} \ frac {c_ {f，n}} {n！} {n！} t^n $是$ 0 $的分析函数，然后让$ c_ {f， n}（x）= \ sum_ {k = 0}^{n} \ binom {n} {n} {k} c_ {f，k} x^{n-k} $是appell polynomials的顺序，称为$ \ textit {c-polynomials of f textit { $ c_ {f，n} $。我们还将$ p_ {f，n}（x）$定义为与函数$ p_ {f}（t）= f（t）（e^t）（e^t-1）/t $相关的c-polynomials的序列，称为$ \ textit {p-polynomials staiment {p-polynomials stocity for f} $。这项工作调查了三个主要主题。首先，我们检查了c-多项式和p多项式的特性以及连接它们的基本特征。其次，汲取灵感从p-多项式的定义并在$ f $上遇到其他条件，我们介绍和研究复杂可变函数$ p_ {f}（s，z，z）= \ sum_ {k = 0}^0}^{+iffty} {+\ infty} \ binom {z} p _} $ s^z $函数，由$ s^{（z，f）} $表示。第三，本文的重大贡献是Hurwitz Zeta功能及其基本属性（最著名的是Hurwitz的公式）的概括，它是通过构建由$ l（z，f）= \ sum_ = \ sum_ {n = n_ {n = n_ {f}}}^{+sum_ {f）{并称为$ \ textIt {与f} $相关的lc命令（常数$ n_ {f} $是一个积极的整数，取决于$ f $的选择）。这项研究提供了与给定的分析功能$ f $相关的c-多项式，p多项式和LC函数的详细分析，彻底检查了它们的相互关系，并引入了具有与Riemann Zeta Zeta Zeta的功能相等的新颖和广泛的LC功能等方程的新颖和广泛的LC功能等方程的研究方向，并介绍了该功能。
Let $f(t)=\sum_{n=0}^{+\infty}\frac{C_{f,n}}{n!}t^n$ be an analytic function at $0$, and let $C_{f, n}(x)=\sum_{k=0}^{n}\binom{n}{k}C_{f,k} x^{n-k}$ be the sequence of Appell polynomials, referred to as $\textit{C-polynomials associated to f}$, constructed from the sequence of coefficients $C_{f,n}$. We also define $P_{f,n}(x)$ as the sequence of C-polynomials associated to the function $p_{f}(t)=f(t)(e^t-1)/t$, called $\textit{P-polynomials associated to f}$. This work investigates three main topics. Firstly, we examine the properties of C-polynomials and P-polynomials and the underlying features that connect them. Secondly, drawing inspiration from the definition of P-polynomials and subject to an additional condition on $f$, we introduce and study the complex-variable function $P_{f}(s,z)=\sum_{k=0}^{+\infty}\binom{z}{k}P_{f,k}s^{z-k}$, which generalizes the $s^z$ function and is denoted by $s^{(z,f)}$. Thirdly, the paper's significant contribution is the generalization of the Hurwitz zeta function and its fundamental properties, most notably Hurwitz's formula, by constructing a novel class of functions defined by $L(z,f)=\sum_{n=n_{f}}^{+\infty}n^{(-z,f)}$, which are intrinsically linked to C-polynomials and referred to as $\textit{LC-functions associated to f}$ (the constant $n_{f}$ is a positive integer dependent on the choice of $f$). This research offers a detailed analysis of C-polynomials, P-polynomials, and LC-functions associated to a given analytic function $f$, thoroughly examining their interrelations and introducing unexplored research directions for a novel and expansive class of LC-functions possessing a functional equation equivalent to that of the Riemann zeta function, thereby highlighting the potential applications and implications of the findings.
