**Theorem 14.** *<sup>m</sup>*

$$\sum\_{n=0}^{m} \sum\_{k=0}^{n} (-1)^{n} \frac{(n)\_{k} a^{n-k} S\_{2}(m, n)}{k + 1} = \sum\_{j=0}^{m} \binom{m}{j} B\_{m-j} B l\_{j} \left( -a \right) \,. \tag{67}$$

By applying the fermionic *p*-adic integral to Equation (58) and combining the final equation with Equations (63) and (66), we also arrive at the following theorem:

### **Theorem 15.** *<sup>m</sup>*

$$\sum\_{n=0}^{m} \sum\_{k=0}^{n} \left(-1\right)^{n} \frac{\left(n\right)\_{k} a^{n-k} S\_{2}\left(m, n\right)}{2^{k}} = \sum\_{j=0}^{m} \binom{m}{j} E\_{m-j} B l\_{j}\left(-a\right) \,. \tag{68}$$

Moreover, by integrating Equation (58) with respect to *x* from 0 to 1, we have:

$$\sum\_{n=0}^{m} a^n S\_2(m, n) \int\_0^1 \mathbb{C}\_n \left( x; a \right) dx = \sum\_{j=0}^m \binom{m}{j} B l\_j \left( -a \right) \int\_0^1 x^{m-j} dx. \tag{69}$$

On the other hand, by integrating Equation (2) with respect to *x* from 0 to 1, we also have:

$$\int\_0^1 \mathbb{C}\_n\left(\mathbf{x}; a\right) d\mathbf{x} = \sum\_{j=0}^n (-1)^{n-j} \binom{n}{j} \frac{1}{a^j} \int\_0^1 \left(\mathbf{x}\right)\_j d\mathbf{x},\tag{70}$$

By making use of the following definition of the well-known Cauchy numbers (or the Bernoulli numbers of the second kind) *bn* (0) (*cf*. [4]):

$$b\_n(0) = \int\_0^1 (\mathbf{x})\_n \, d\mathbf{x}\_\prime \tag{71}$$

Equation (70) yields:

$$\int\_0^1 \mathbb{C}\_n\left(\mathbf{x}; a\right) d\mathbf{x} = \sum\_{j=0}^n \left(-1\right)^{n-j} \binom{n}{j} \frac{b\_j\left(0\right)}{a^j}.\tag{72}$$

Combining the above equation with Equation (69), we arrive at the following theorem:

**Theorem 16.** *<sup>m</sup>*

$$\sum\_{n=0}^{m} \sum\_{j=0}^{n} (-1)^{n-j} \binom{n}{j} a^{n-j} \mathbb{S}\_2(m, n) b\_j(0) = \sum\_{j=0}^{m} \binom{m}{j} \frac{Bl\_j(-a)}{m-j+1}.\tag{73}$$

#### **6. Applications in the Probability Distribution Function**

In this section, we investigate some applications of the numbers *<sup>Y</sup>*(−*k*) *<sup>n</sup>* (*λ*). Assume that 0 <sup>&</sup>lt; *<sup>p</sup>* <sup>≤</sup> <sup>1</sup> and *n* = 0, 1, 2, . . . , *k*. We set the following discrete probability distribution:

$$f\left(p;k,n\right) = \frac{\left(-1\right)^{k-n}2^k}{n!p^n}Y\_n^{(-k)}\left(p\right) \tag{74}$$

where *p* is a probability of success, *k* is number of trials, *n* is number of successes in *k* trials, and *n* = 0, 1, 2, . . . , *k*. Therefore, *f* (*p*; *k*, *n*) is binomially distributed with parameters (*k*, *p*).

*Properties of Discrete Probability Distribution f* (*p*; *k*, *n*)

Here, we give some properties of discrete probability distribution *f* (*p*; *k*, *n*). We examine the properties of the probability distribution *f* (*p*; *k*, *n*) with a random variable with parameters *k*, *n*, and *p* as follows:

For all *k*, *n*, *p* with 0 ≤ *n* ≤ *k* and 0 < *p* ≤ 1, 0 ≤ *f* (*p*; *k*, *n*) ≤ 1. That is *f* (*p*; *k*, *n*) ≥ 0. The probability distribution function *f* (*p*; *k*, *n*) satisfies that:

$$\sum\_{n=0}^{\infty} f\left(p; k, n\right) = 1.$$

Computing the distribution function *f* (*p*; *k*, *n*). Suppose that *X* is a binomial with parameters (*k*, *p*). To computing its distribution function:

$$P(X \le j) = \sum\_{n=0}^{j} f\left(p; k, n\right),$$

where *j* = 0, 1,. . . ,*k*.

In order to compute its expected value and variance for random variable with parameters *k* and *p*:

$$E\left[X^{\overline{v}}\right] = \sum\_{n=0}^{k} n^{v} f\left(p; k, n\right) \tag{75}$$

Observe that the probability distribution function *f* (*p*; *k*, *n*) is a modification of the binomial probability distribution function with parameters (*k*, *p*). Substituting *v* = 1 into Equation (75), *E* [*X*] = *kp*. Substituting *v* = 2 into Equation (75), variance *E X*2 <sup>−</sup> (*<sup>E</sup>* [*X*])<sup>2</sup> <sup>=</sup> *kp* (<sup>1</sup> <sup>−</sup> *<sup>p</sup>*).

If we take *k* → ∞, then the distribution *f* (*p*; *k*, *n*) goes to the Poisson distribution. On the other hand the Poisson–Charlier polynomials are orthogonal with respect to the Poisson distribution (*cf*. [18,24]).

#### **7. Conclusions**

Applications of generating functions are used in many areas, and we used them to study new families of combinatorial numbers and polynomials. We then studied properties of these new families, which yielded a handful of new identities and relations. Namely, these identities were related to numerous special numbers, special polynomials, and special functions such as the Bersntein basis functions, the Stirling numbers, the Bell polynomials (or exponential polynomials), the Poisson–Charlier polynomials, and the probability distribution functions. Furthermore, we should note that newly defined combinatorial numbers in this paper gave a different approach to the binomial (or Newton) distribution and the Poisson distribution, as well as combinatorial sums including the Bernoulli numbers, the Euler numbers, the Cauchy numbers (or the Bernoulli numbers of the second kind), and combinatorial numbers. This is why the results of this paper have the potential to be used in numerous areas such as mathematics, probability, physics, and in other associated areas.

**Author Contributions:** Investigation, I.K., B.S., Y.S.; wirting-original draft, I.K., B.S., Y.S.; writing-review and editing, I.K., B.S., Y.S.

**Funding:** This research received no external funding.

**Acknowledgments:** This paper is dedicated to Hari Mohan Srivastava on the occasion of his 80th Birthday. Yilmaz Simsek was supported by the Scientific Research Project Administration of Akdeniz University.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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