# Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function

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## Abstract

**:**

## 1. Introduction

## 2. New Families of the Combinatorial Numbers and Polynomials

**Theorem**

**1.**

**Theorem**

**2.**

**Corollary**

**1.**

**Theorem**

**3.**

**Corollary**

**2.**

**Corollary**

**3.**

**Corollary**

**4.**

## 3. Derivative Formulas and Recurrence Relations Arising from Differential Equations of Generating Functions

**Theorem**

**4.**

**Theorem**

**5.**

**Theorem**

**6.**

**Theorem**

**7.**

**Theorem**

**8.**

**Remark**

**1.**

**Theorem**

**9.**

## 4. Some Identities and Relations Derived from Functional Equations of Generating Functions

**Theorem**

**10.**

**Theorem**

**11.**

**Theorem**

**12.**

**Theorem**

**13.**

## 5. Some Identities and Relations Arising from the p-adic Integrals and Riemann Integral

**Theorem**

**14.**

**Theorem**

**15.**

**Theorem**

**16.**

## 6. Applications in the Probability Distribution Function

#### Properties of Discrete Probability Distribution $f\left(\right)open="("\; close=")">p;k,n$

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Kucukoglu, I.; Simsek, B.; Simsek, Y.
Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function. *Axioms* **2019**, *8*, 112.
https://doi.org/10.3390/axioms8040112

**AMA Style**

Kucukoglu I, Simsek B, Simsek Y.
Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function. *Axioms*. 2019; 8(4):112.
https://doi.org/10.3390/axioms8040112

**Chicago/Turabian Style**

Kucukoglu, Irem, Burcin Simsek, and Yilmaz Simsek.
2019. "Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function" *Axioms* 8, no. 4: 112.
https://doi.org/10.3390/axioms8040112