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Limiting Values and Functional and Difference Equations^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Lerch Zeta-Function

## 3. Limit Values in Riemann’s Fragment II

**Definition**

**1.**

**Theorem**

**1**

**.**Let $\xi =\frac{M}{Q}$ be a rational number with M even and $Q>1$ and let $z=y{e}^{\pi i\xi},\phantom{\rule{0.277778em}{0ex}}y\in [0,1)$. Then we have

**Proposition**

**1.**

**Theorem**

**2**

**.**Let M be a fixed modulus $>1$. Let ${R}_{n}\left(x\right)$ denote a complex-valued function defined on $\mathrm{I}=[0,1]$ such that ${R}_{n}\left(x\right)={R}_{k}\left(x\right)$ for $n\equiv k\phantom{\rule{0.166667em}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}q)$, and a fortiori, there are M different functions. Assume that each ${R}_{k}\left(x\right)$ is of Lipschitz $\alpha ,\phantom{\rule{0.277778em}{0ex}}\alpha \ge 1$, ${R}_{k}\left(x\right)\in Lip\phantom{\rule{0.277778em}{0ex}}\alpha $ and $\sum _{k=1}^{M}}{R}_{k}\left(x\right)=0$ for each $x\in \mathrm{I}$. Then the Dirichlet series

**Example**

**1.**

**Example**

**2**

**.**We let

## 4. Generalized Euler Constants

**Theorem**

**3**

**.**The Laurent expansion

**Proposition**

**2**

**.**For $\tau \in \mathcal{H}$ and $n\in \mathbb{N}$

**Theorem**

**4**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2**

**.**

**Example**

**3**

**.**

## 5. Difference Equations

**Theorem**

**5**

**.**

## 6. Abel-Tauber Process

**Lemma**

**1**

**.**For $\left\{{a}_{n}\right\}$ and $A\left(x\right)$ as above suppose $g\left(t\right)$ is of class ${C}^{1}$ on $[{\lambda}_{1},\infty )$. then the formula for integration by parts

**Lemma**

**2**

**.**Suppose that the functional equation

**Theorem**

**6**

**.**Suppose

**Proof.**

**Corollary**

**3**

**.**If ${\lambda}_{n}=n$ and

**Remark**

**1.**

## 7. Quellenangaben

## 8. Elucidation of Some Identities

#### 8.1. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Wang, N.-L.; Agarwal, P.; Kanemitsu, S.
Limiting Values and Functional and Difference Equations. *Mathematics* **2020**, *8*, 407.
https://doi.org/10.3390/math8030407

**AMA Style**

Wang N-L, Agarwal P, Kanemitsu S.
Limiting Values and Functional and Difference Equations. *Mathematics*. 2020; 8(3):407.
https://doi.org/10.3390/math8030407

**Chicago/Turabian Style**

Wang, N.-L., Praveen Agarwal, and S. Kanemitsu.
2020. "Limiting Values and Functional and Difference Equations" *Mathematics* 8, no. 3: 407.
https://doi.org/10.3390/math8030407