# 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

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**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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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