# 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

- Debnath, L.; Bhatta, D. Integral Transforms And Their Applications, 3rd ed.; CRC Press: Boca Raton, FL, USA, 2015. [Google Scholar]
- Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions; D. Reidel Publishing Company: Dordrecht, The Netherlands, 1974. [Google Scholar]
- Graham, R.L.; Knuth, D.E.; Patashnik, O. Concrete Mathematics: A Foundation for Computer Science; Addison-Wesley: Boston, MA, USA, 1994. [Google Scholar]
- Stanley, R.P. Enumerative Combinatorics, Volume 2; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Boros, G.; Moll, V.H. Irresistible Integrals: Symbolics, Analysis and Experiements in the Evaluation of Integrals; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Flajolet, P.; Sedgewick, R. Analytic Combinatorics; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Gradshetyn, I.S.; Rhyzhik, I.M. Table of Integrals, Series, and Products; Academic Press: Cambridge, MA, USA, 1994. [Google Scholar]
- Borwein, D.; Borwein, J.M.; Girgensohn, R. Explicit evaluation of Euler sums. Proc. Edinb. Math. Soc.
**1995**, 38, 277–294. [Google Scholar] [CrossRef] [Green Version] - Milgram, M.S. Integral and Series Representations of Riemann’s Zeta Function, Dirichlet’s eta Function and A Medley of Related Results. arXiv
**2012**, arXiv:1208.3429. [Google Scholar] [CrossRef] - Schmidt, M.D. Zeta Series Generating Function Transformations Related to Generalized Stirling Numbers and Partial Sums of the Hurwitz Zeta Function. Online J. Anal. Comb.
**2018**, 13, 1–31. [Google Scholar] - Schmidt, M.D. Zeta series generating function transformztions related to polylogarithm functions and the k–order harmonic numbers. Online J. Anal. Comb.
**2017**, 12, 1–22. [Google Scholar] - Schmidt, M.D. Square series generating function transformations. Inequalities Spec. Funct.
**2017**, arXiv:1609.02803. [Google Scholar] - Granovskya, B.L.; Starkb, D.; Erlihson, M. Meinardus’ theorem on weighted partitions: Extensions and a probabilistic proof. Adv. Appl. Math.
**2008**, 41, 307–328. [Google Scholar] [CrossRef] - Bernstein, M.; Sloane, N.J.A. Some Cannonical Sequences of Integers. arXiv
**2002**, arXiv:math/0205301. [Google Scholar] - Andrews, G.E. q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra; American Mathematical Society: Providence, RI, USA, 1986. [Google Scholar]
- Olver, F.W.J.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W. (Eds.) NIST Handbook of Mathematical Functions; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Remmert, R. Classical Topics in Complex Function Theory; Springer: Berlin/Heidelberg, Germany, 1991. [Google Scholar]
- Fekih-Ahmed, L. On the power series expansions of the reciprocal gamma function. arXiv
**2017**, arXiv:abs/1407.5983. [Google Scholar] - Tolstov, G.P. Fourier Series; Dover Publications: Mineola, NY, USA, 2014. [Google Scholar]
- An Integral Formula For The Reciprocal Gamma Function. Available online: https://math.stackexchange.com/questions/2274972/an-integral-formula-for-the-reciprocal-gamma-function (accessed on 22 May 2018).
- Schmidt, M.D. Generalized j–factorial functions, polynomials, and applications. J. Integer Seq.
**2010**, 13, 1–54. [Google Scholar]

**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.

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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