# Fractional-Modified Bessel Function of the First Kind of Integer Order

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*Mathematics*: 10th Anniversary)

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- Let spanish $\alpha \in (0,1]$ and $C\in \mathbb{R}$, then: ${D}_{t}^{\alpha}(C)=0$;
- Let $\alpha \in (0,1]$ and $r>0$, then ${D}_{t}^{\alpha}({t}^{r})=\frac{\mathrm{\Gamma}(r+1)}{\mathrm{\Gamma}(r-\alpha +1)}{t}^{r-\alpha}$;
- Let $\gamma \in [0,1)$ and $r>-1$, then ${\mathcal{I}}^{\gamma}({t}^{r})=\frac{\mathrm{\Gamma}(r+1)}{\mathrm{\Gamma}(r+\gamma +1)}{t}^{r+\gamma};$
- Let $\gamma \in (0,1)$ and $r>0$, then ${D}_{t}^{\gamma}({\mathcal{I}}^{\gamma}({t}^{r}))={\mathcal{I}}^{\gamma}({D}_{t}^{\gamma}({t}^{r}))={t}^{r}$.

## 3. Fractional Communicability in Graphs

## 4. Fractional Communicabilities in Path and Cycle Graphs

**Definition 1.**

**Remark 1.**

**Remark 2.**

**Theorem 1.**

**Proof.**

**Theorem 2.**

**Example 1.**

## 5. On the Estrada–Mittag–Leffler Indices of ${\mathit{P}}_{\mathit{n}}$ and ${\mathit{C}}_{\mathit{n}}$

## 6. Power Series of the the FMBF of the First Kind

**Lemma 1.**

**Proof.**

**Remark 3.**

**Lemma 2.**

**Proof.**

**Remark 4.**

**Remark 5.**

## 7. Differential Properties of the FMBF of the First Kind

**Theorem 3.**

- $${D}_{z}^{\alpha}\left({\mathcal{E}}_{\nu ,\alpha}\left({z}^{\alpha}\right)\right)={\mathcal{E}}_{\nu -1,\alpha}\left({z}^{\alpha}\right)-\alpha \nu \xb7{\mathcal{I}}^{1-\alpha}({z}^{-1}\xb7{\mathcal{E}}_{\nu ,\alpha}\left({z}^{\alpha}\right))$$
- $${D}_{z}^{\alpha}\left({\mathcal{E}}_{\nu ,\alpha}\left({z}^{\alpha}\right)\right)={\mathcal{E}}_{\nu +1,\alpha}\left({z}^{\alpha}\right)+\alpha \nu \xb7{\mathcal{I}}^{1-\alpha}({z}^{-1}\xb7{\mathcal{E}}_{\nu ,\alpha}\left({z}^{\alpha}\right))$$
- $${D}_{z}^{\alpha}\left({\mathcal{E}}_{\nu ,\alpha}\left({z}^{\alpha}\right)\right)={\displaystyle \frac{1}{2}}({\mathcal{E}}_{\nu -1,\alpha}\left({z}^{\alpha}\right)+{\mathcal{E}}_{\nu +1,\alpha}\left({z}^{\alpha}\right)).$$

**Lemma 3.**

**Proof.**

**Proof.**

**Remark 6.**

## 8. Open Problems

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Lebedev, N.N. Special Functiones and Their Applications; Dover Pub. Inc.: New York, NY, USA, 1972; p. 108. [Google Scholar]
- Bowman, F. Introduction to Bessel Functions; Dover Pub. Inc.: New York, NY, USA, 1958; p. 41. [Google Scholar]
- Watson, G.N. A Treatise on the Theory of Bessel Functions; Cambridge Mathematical Library: New York, NY, USA, 1995; p. 77. [Google Scholar]
- Simpson, H.; Spector, S. Some monotonicity results for ratios of modified Bessel functions. J. Inequal. Appl.
**1984**, 42, 95–98. [Google Scholar] [CrossRef][Green Version] - Hwang, Y. Difference-Based Image Noise Modeling Using Skellam Distribution. IEEE Trans. Pattern Anal. Mach. Intell.
**2012**, 34, 1329–1341. [Google Scholar] [CrossRef] [PubMed] - Wolfe, P.; Hirakawa, K. Efficient Multivariate Skellam Shrinkage for Denoising Photon-Limited Image Data: An Empirical Bayes Approach. In Proceedings of the 16th IEEE International Conference on Image Processing (ICIP-09), Cairo, Egypt, 7–10 November 2009; pp. 2961–2964. [Google Scholar]
- Karlis, D.; Ntzoufras, I. Analysis of sports data using bivariate Poisson models. J. R. Stat. Soc. Ser. D
**2003**, 52, 381–393. [Google Scholar] [CrossRef] - Robert, C. Modified Bessel functions and their applications in probability and statistics. Stat. Probab. Lett.
**1990**, 9, 155–161. [Google Scholar] [CrossRef] - Gutman, I.; Graovac, A. Estrada index of cycles and paths. Chem. Phys. Lett.
**2007**, 436, 294–296. [Google Scholar] [CrossRef] - Gaunt, R.E. Bounds for an integral of the modified Bessel function of the first kind and expressions involving it. J. Math. Anal. Appl.
**2021**, 502, 125216. [Google Scholar] [CrossRef] - Baricz, Á. Powers of modified Bessel functions of the first kind. Appl. Math. Lett.
**2010**, 23, 722–724. [Google Scholar] [CrossRef][Green Version] - Yang, Z.-H.; Tian, J.-F.; Zhu, Y.-R. New sharp bounds for the modified Bessel function of the first kind and Toader-Qi mean. Mathematics
**2020**, 8, 901. [Google Scholar] [CrossRef] - Baricz, Á. Functional inequalities involving Bessel and modified Bessel functions of the first kind. Expo. Math.
**2008**, 26, 279–293. [Google Scholar] [CrossRef][Green Version] - Baricz, Á.; Kupán, P.; Szász, R. The radius of starlikeness of normalized Bessel functions of the first kind. Proc. Am. Math. Soc.
**2014**, 142, 2019–2025. [Google Scholar] [CrossRef][Green Version] - Deleaval, L.; Demni, N. On a Neumann-type series for modified Bessel functions of the first kind. Proc. Am. Math. Soc.
**2018**, 146, 2149–2161. [Google Scholar] [CrossRef][Green Version] - Abbas, M.I.; Ragusa, M.A. Nonlinear fractional differential inclusions with non-singular Mittag-Leffler kernel. AIMS Math.
**2022**, 7, 20328–20340. [Google Scholar] [CrossRef] - Van, A.V.; Jagdev, S.; Tuan, N.A. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Elec. Res. Arch.
**2021**, 29, 3581–3607. [Google Scholar] - Nasir, J.; Qaisar, S.; Butt, S.I.; Khan, K.A.; Mabela, R.M. Some Simpson’s Riemann–Liouville fractional integral inequalities with applications to special functions. J. Funct. Spaces
**2022**, 2022, 2113742. [Google Scholar] [CrossRef] - Estrada, E.; Hatano, N. Communicability in complex networks. Phys. Rev. E
**2008**, 77, 036111. [Google Scholar] [CrossRef][Green Version] - Estrada, E.; Hatano, N.; Benzi, M. The physics of communicability in complex networks. Phys. Rep.
**2012**, 514, 89–119. [Google Scholar] [CrossRef][Green Version] - Estrada, E.; Rodriguez-Velazquez, J.A. Subgraph centrality in complex networks. Phys. Rev. E
**2005**, 71, 056103. [Google Scholar] [CrossRef][Green Version] - Galué, L. A generalized Bessel function. Integr. Spec. Funct.
**2003**, 14, 395–401. [Google Scholar] [CrossRef] - Milici, C.; Drăgănescu, G.; Tenreiro Machado, J. Introduction to Fractioanl Differential Equations; Spinger: Berlin/Heidelberg, Germany, 2019. [Google Scholar]
- Abadias, L.; Estrada-Rodriguez, G.; Estrada, E. Fractional-order susceptible-infected model: Definition and applications to the study of COVID-19 main protease. Fract. Calc. Appl. Anal.
**2020**, 23, 635–655. [Google Scholar] [CrossRef] [PubMed] - Arrigo, F.; Durastante, F. Mittag-Leffler Functions and their Applications in Network Science. SIAM J. Matrix Anal. Appl.
**2021**, 42, 1581–1601. [Google Scholar] [CrossRef] - Estrada, E. The many facets of the Estrada indices of graphs and networks. SeMA J.
**2022**, 79, 57–125. [Google Scholar] [CrossRef]

**Figure 1.**Plot of the functions ${\mathcal{E}}_{\nu ,\alpha}(z)$ for $z\in \mathbb{Z}$ and for $\nu =0$ (

**a**) and $\nu =1$ (

**b**) as well as for different values of the fractional parameter $\alpha $. The functions were computed using numerical integration on the basis of Equation (21).

**Table 1.**Computational results of values of ${({E}_{\alpha}({P}_{20}))}_{v,v}$ and ${({E}_{\alpha}({C}_{40}))}_{v,v}$ computed using the Mittag–Leffler matrix function and of its approximate values using the FMBF of the first kind, ${({\widehat{E}}_{\alpha}({P}_{n}))}_{v,v}$ and ${\widehat{E}}_{\alpha}({C}_{40})$ for which we have used numerical integration.

v | ${({\mathit{E}}_{\mathit{\alpha}}({\mathit{P}}_{20}))}_{\mathit{v},\mathit{v}}$ | ${({\widehat{\mathit{E}}}_{\mathit{\alpha}}({\mathit{P}}_{20}))}_{\mathit{v},\mathit{v}}$ | ||||
---|---|---|---|---|---|---|

$\mathbf{\alpha}=\mathbf{0}.\mathbf{4}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{6}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{8}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{4}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{6}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{8}$ | |

1 | 14.0351209 | 3.1971466 | 2.0259468 | 14.0351209 | 3.1971466 | 2.0259468 |

2 | 40.2363816 | 6.2883718 | 3.2379302 | 40.2363816 | 6.2883718 | 3.2379302 |

3 | 61.4983128 | 7.3852388 | 3.4402925 | 61.4983128 | 7.3852388 | 3.4402925 |

v | ${({E}_{\alpha}({C}_{40}))}_{v,v}$ | ${({\widehat{E}}_{\alpha}({C}_{40}))}_{v,v}$ | ||||

$\alpha =0.4$ | $\alpha =0.6$ | $\alpha =0.8$ | $\alpha =0.4$ | $\alpha =0.6$ | $\alpha =0.8$ | |

1 | 80.9762993 | 7.6594588 | 3.4584489 | 80.9762993 | 7.6594588 | 3.4584489 |

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**MDPI and ACS Style**

Martín, A.; Estrada, E.
Fractional-Modified Bessel Function of the First Kind of Integer Order. *Mathematics* **2023**, *11*, 1630.
https://doi.org/10.3390/math11071630

**AMA Style**

Martín A, Estrada E.
Fractional-Modified Bessel Function of the First Kind of Integer Order. *Mathematics*. 2023; 11(7):1630.
https://doi.org/10.3390/math11071630

**Chicago/Turabian Style**

Martín, Andrés, and Ernesto Estrada.
2023. "Fractional-Modified Bessel Function of the First Kind of Integer Order" *Mathematics* 11, no. 7: 1630.
https://doi.org/10.3390/math11071630