An Application of Pascal Distribution Series on Certain Analytic Functions Associated with Stirling Numbers and S˘al˘agean Operator

In the present paper, we will observe that the S˘al˘agean diﬀerential operator can be written in terms of Stirling numbers. Furthermore, we ﬁnd a necessary and suﬃcient condition and inclusion relation for Pascal distribution series to be in the class P k ( λ , α ) of analytic functions with negative coeﬃcients deﬁned by the S˘al˘agean diﬀerential operator. Also, we consider an integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.


Preliminaries
Special functions are used in many applications of physics, engineering, and applied mathematics and statistics. Special polynomials have a close connection with number theory, and one of the most important sets of special numbers is the class of Stirling numbers (of the first and second kind), introduced in 1730 by the Scottish mathematician James Stirling.
In combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of k objects into j nonempty subsets and is denoted by S(k, j) or by b k,j as used in this paper. ese numbers occur in the field of mathematics called combinatorics and the study of partitions. In this paper, we will observe that the Sȃlȃgean differential operator D k can be written in terms of Stirling numbers. Let Furthermore, let T be a subclass of A consisting of functions of the form f(z) � z − ∞ n�2 a n z n , z ∈ D. (2) For a function f(z) in A, we define and in general, we have e differential operator D k was introduced by Sȃlȃgean [1].
For example, where (ii) If k � 3, we have where where (iv) If k � 5, we have where Table 1 represents the coefficients b kj of z k f (k) (z). Table 1 (see [2]) shows the first few possibilities for Stirling numbers of the second kind. Also, from this table, we note that: [n(n − 1) + n]a n z n , and + 25n(n − 1)(n − 2) + 15n(n − 1) + n]a n z n .
From (16)- (19), we conclude that In general, we have For functions f ∈ A given by (1) and g ∈ A given by g(z) � z + ∞ n�2 b n z n , we recall that the well-known Hadamard product of f and g is given by For e function class R ϵ (C, D) was introduced by Dixit and Pal [3].
With the help of the differential operator D k , we say that a function f(z) belonging to A is said to be in the class , and for all z in D. Furthermore, we define the class P k (λ, α) by e class P T (λ, α) was introduced and studied by Aouf and Srivatava [4].
We note that, by specializing the parameters k and λ, we obtain the following subclasses: T * (α) and C(α) represent the classes of starlike functions of order α and convex functions of order α with negative coefficients, respectively, introduced and studied by Silverman [5] In statistics and probability, distributions of random variables play a basic role and are used extensively to describe and model a lot of real-life phenomenon; they describe the distribution of the probabilities over the values of the random variable. In recent years, many researchers have examined some important features in the geometric function theory, such as coefficient estimates, inclusion relations, and conditions of being in some known classes, using different probability distributions such as the Poisson, Pascal, Borel, Mittag-Leffler-type Poisson distribution, etc.(see, for example, [6][7][8][9][10]). e probability density function of a discrete random variable X which follows the Pascal distribution is given by Very recently, El-Deeb et al. [11] introduced a power series whose coefficients are probabilities of the Pascal distribution where m ≥ 1 and 0 ≤ σ ≤ 1 and we note that, by a ratio test, the radius of convergence of above series is infinity. We also define the series Now, we considered the linear operator defined by the Hadamard product Motivated by several earlier results on connections between various subclasses of analytic and univalent functions, using hypergeometric functions, generalized Bessel functions, Struve functions, Poisson distribution series, and Pascal distribution series (see, for example, [12], [13][14][15], [7][8][9][16][17][18][19][20][21][22][23], [24]), we determine a necessary and sufficient condition for Υ m σ (z) to be in our class P k (λ, α). Furthermore, we give sufficient conditions for e following results will be required in our investigation.
e result is sharp for the function
First of all, with the help of Lemma 1, we obtain the following necessary and sufficient condition for Υ m σ (z) to be in P k (λ, α).
Proof. In view of Lemma 1, we only need to show that Using (22) and (38), we have

Journal of Function Spaces
(41) erefore, we see that the last expression is bounded above by 1 − α if (39) is satisfied.

Inclusion Properties
Making use of Lemma 2, we will study the action of the Pascal distribution series on the class P k (λ, α).
Proof. In view of Lemma 1, it suffices to show that L ≤ 1 − α, where 8 Journal of Function Spaces (43) Applying Lemma 2, we find from equations (22) and (38) that

Journal of Function Spaces
However, this last expression is bounded by 1 − α, if (42) holds. is completes the proof of eorem 2.

An Integral Operator
In this section, we consider the integral operator G m σ defined by Proof. From definitions (31) and (45), it is easily verified that en, by Lemma 1, we need only to show that or equivalently, e remaining part of the proof of eorem 3 is similar to that of eorem 2, and so, we omit the details.

Corollaries and Consequences
By specializing the parameter λ � 1 in eorems 1-3, we obtain the following corollaries. (52) Corollary 2. Let k ≥ 2 and f ∈ R ϵ (C, D). en, (53) (54) Corollary 6. e integral operator G m σ f(z) defined by (45) is in the class P 2 (λ, α) if and only if Remark 1. Using relation (22) and Lemma 1, we can obtain new necessary and sufficient conditions and inclusion relations for the Pascal distribution series to be in the class P k (λ, α) for k � 3, 4, . . ..

Conclusions
e Sȃlȃgean differential operator plays an important role in the geometric function theory. Several authors have used this operator to define and consider the properties of certain known and new classes of analytic univalent functions (see, for example, [25,26]). In the present paper, and due to the earlier works (see, for example, [11,16,18]), we find a necessary and sufficient condition and inclusion relation for the Pascal distribution series to be in the class P k (λ, α) of analytic functions associated with the Stirling numbers and Sȃlȃgean differential operator. Furthermore, we consider an integral operator related to the Pascal distribution series. Some interesting corollaries and applications of the results are also discussed. Making use of the relation (22) could inspire researchers to find new necessary and sufficient conditions and inclusion relations for the Pascal distribution series to be in different classes of analytic functions with negative coefficients defined by the Sȃlȃgean differential operator.

Data Availability
No data were used to support this study. 12 Journal of Function Spaces

Conflicts of Interest
e author declares that there are no conflicts of interest.

Authors' Contributions
e author read and approved the final manuscript.