# A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions

## Abstract

**:**

## 1. Introduction

#### 1.1. Definitions

#### 1.2. From Hobby To Short Note: OGF-to-EGF Conversion Formulas

#### 1.3. Examples: Integral transformations of a Sequence Generating Function

#### 1.4. Results Proved in This Note

**Theorem**

**1**(OGF-to-EGF Integral Formula I)

**.**

**Theorem**

**2**(OGF-to-EGF Integral Formula II)

**.**

## 2. Integral Representations of the Reciprocal Gamma Function

#### 2.1. The Hankel Loop Contour for the Reciprocal Gamma Function

**Lemma**

**1.**

**Proof.**

**Proof**

**of**

**Theorem**

**1.**

#### 2.2. Examples: Applications of the Integral Formula on the Real Line

## 3. An Integral Formula from Fourier Analysis

**Proof**

**of**

**Theorem**

**2.**

**Alternate**

**Proof**

**of**

**Theorem**

**2.**

**Remark**

**1**(Generalizations of series expansions from Fourier series)

**.**

#### Examples: Generalizations and Solutions to a Long-Standing Forum Post

## 4. Concluding Remarks

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The Hankel loop contour providing an integral representation of the reciprocal gamma function when $Re\left(z\right)>0$. This contour starts positively from the right, traverses the horizontal line ${L}_{\infty}^{+}(\delta ,\epsilon )$ at distance $+\delta $ from the x-axis from $+\infty \to \sqrt{|{\epsilon}^{2}-{\delta}^{2}|}$, then enters the semi-circular loop about the origin of radius $\epsilon $ denoted by ${C}_{\epsilon}\left(\delta \right)$ at the point ${P}_{1}$, and then at the point ${P}_{2}=(\sqrt{|{\epsilon}^{2}-{\delta}^{2}|},-\delta )$ traverses the last horizontal line ${L}_{\infty}^{-}(\delta ,\epsilon )$ back to infinity parallel to the x-axis.

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Schmidt, M.D. A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions. *Axioms* **2019**, *8*, 62.
https://doi.org/10.3390/axioms8020062

**AMA Style**

Schmidt MD. A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions. *Axioms*. 2019; 8(2):62.
https://doi.org/10.3390/axioms8020062

**Chicago/Turabian Style**

Schmidt, Maxie D. 2019. "A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions" *Axioms* 8, no. 2: 62.
https://doi.org/10.3390/axioms8020062