Abstract
The paper aims to introduce and investigate a new class of generalized Fubini-type polynomials. The generating functions, special cases, and properties are introduced. Using the generating functions, various interesting identities, and relations are derived. Also, special polynomials are obtained from the general class of polynomials. Finally, probabilistic applications and some probabilistic properties are obtained.
1. Introduction
In recent years, Fubini-type polynomials have gained attention not only in combinatorial mathematics but also in applications to differential equations and probability theory; so many authors have studied its properties and obtained important results [1–4]. For example, the classical Fubini polynomials are defined by (see [4]).where is the Stirling numbers of the second kind [5], These polynomials satisfy the following generating function:
Putting in (1), we obtain the Fubini number or the ordered Bell number , defined by
The bivariate Fubini polynomials of order are generalization of the classical Fubini polynomials and they are defined by the following generating function:
Putting , , the two-variable Fubini polynomials are obtained by
Also, putting in (4), we obtainwhere and are called the higher order Fubini polynomials and the higher order Fubini numbers, respectively, (see [6, 7]). Also, Kilar and Simsek [2] obtained serval 2-variable special polynomials from the family of 2-variable general polynomials , for more details see ([8–11]).
Additionally, for any , the degenerate exponential functions are defined by
From (7), we getwhere
The degenerate Stirling numbers of the first kind are defined byand the degenerate Stirling numbers of the second kind are given by
The Bell polynomials are defined by the following equation:
Putting , are called the Bell numbers. From (11), we get
Also, the degenerate Bell polynomials are defined by (see [13]).
In recent years, a lot of research on multivariable and generalized polynomials has been introduced. So, we recall the following 2-variable form of special polynomials.
The 2-variable general polynomials (2VGP) [14] are defined bywhere
The 2VGP family contains a number of significant 2-variable special polynomials. Based on suitable selection for the function , different members belonging to the family of 2-variable general polynomials can be obtained, see Table 1.
The aim of this paper is to study a new family of generalized Fubini-type polynomials and to obtain an application in probability theory. For this, we derive an explicit expression, new identities and relations with important polynomials and numbers. Furthermore, we introduce a new three-parameter discrete distribution which may be useful for modeling count data. Submodels such as Bell distribution and Bell–Touchard distribution are obtained, which have an important role in count data (for more details, see([16–18])).
The article is organized as follows: In Section 2, the generalized Fubini-type polynomials are introduced. The generating functions and some special cases for new families of polynomials are introduced. In Section 3, new definitions and certain important relations for new families of polynomials are derived. In Section 4, special polynomials related to the new families of polynomials are considered. In Section 5, the probabilistic application and some probabilistic properties are obtained.
2. Generalized Fubini Apostol-Type Polynomials
Definition 1. The multivariable generalized polynomials are defined bywhere
Definition 2. Let , and be sequences of complex numbers, then the generalized Fubini Apostol-type polynomials are defined as the following:where
Remark 1. Putting and in (19), we obtain the new Fubini Apostol-type numbers, as shown in the following equation:Setting and in (19), gives the following definition.
Definition 3.
Remark 2. If we put in equation (22), then, we obtain the new Apostol-type polynomials, as shown in the following equation:By setting appropriate values for parameters in (19) and (22), we obtain the generating functions and some results for the mixed special polynomials related to . The generating function and some definitions for these polynomials are introduced in Table 2.
The series definition of are obtained by the next theorem:
Theorem 1. The generalized Fubini Apostol-type polynomials is defined by the series:
Proof. Using equation (19) and by using Cauchy-product rule, yields (2.8).
3. Identities and Relations
Theorem 2. Let and . Then,
Proof. From (19)by using Cauchy-product rule, yields (25).
Corollary 1. Setting in (25), we obtain the recurrence relation
Theorem 3. Let , then, for and . The following identity holds true:
Proof. Changing by in (19)by using Cauchy-product rule, yields (29).
Theorem 4. For , we have the relationshipbetween the generalized Fubini Apostol-type polynomials and Stirling number of second kind.
Proof. Using (28) and from definition of Stirling number of second kind, see [28, 29], we haveBy using Cauchy-product rule, yields (31).
Theorem 5. For and . The following identity for holds true:
Proof. changing by in equation (19) and using the following rule [30]in the left hand side becomesRewriting (35) asReplacing by and equating the resulting equation to the previous equation, we obtainOn expanding exponential function in (37) we obtainUsing (34) in the left hand side of (38), we getPutting by , by and by using Cauchy-product rule in the left hand side of (39), we getFinally, comparing the similar coefficients of and in the previous equation yields (33).
Theorem 6. For and , we have the relationshipbetween the generalized Fubini Apostol-type polynomials, generalized Stirling number of first kind and generalized Hermite polynomials.
Proof. Setting in equation (22), from the generating function of generalized Stirling numbers and from the generating function of generalized Hermite polynomials, we haveBy comparing the similar coefficients of in the previous equation, yields (41).
Theorem 7. For and , we have the relationshipbetween the generalized Fubini Apostol-type polynomia, degenerate Stirling number of second kind and generalized Hermite polynomials.
Proof. Putting in equation (22), from the generating function of degenerate generalized Fubini polynomials, see ([26]) and from the generating function of Hermite polynomials, we obtainBy comparing the coefficient of in the previous (44), we obtainSinceTherefore, we get (43).
Theorem 8. For and , we have the relationshipbetween the generalized Fubini Apostol-type polynomials, degenerate Bell polynomials and generalized Hermite polynomials.
Proof. Putting in equation (22) and using equation (14), we obtain (47).
Corollary 2. Setting in (47), we obtain relation between generalized Apostol-type polynomials, Bell polynomials and generalized Hermite polynomials
Theorem 9. For , we have the relationshipbetween the generalized Apostol-type numbers and Bell polynomials.
Proof. Putting in equation (23) and using equation (14), we obtain (49).
Corollary 3. Setting in (49), we obtain relation between generalized Apostol numbers and Bell polynomials.
4. Examples
Selecting appropriate choice for in (22), the new special families of polynomials are obtained from .
4.1. Gould-Hopper Fubini Apostol Type Polynomials
Setting in (22), we obtain Gould–Hopper Fubini Apostol-type polynomials (GHFATP), denoted by which are defined by
We can obtain some definitions and some results for the GHFATP, these results are listed in Table 3.
4.2. Laguerre Fubini Apostol-Type Polynomials
Setting in (22), gives Laguerre Fubini Apostol-type polynomials (LFATP), denoted by which are defined by
We can obtain some definitions and some results for the LFATP, these results are listed in Table 4.
4.3. Truncated Exponential Fubini Apostol-Type Polynomials
Setting in (22), gives Truncated exponential Fubini Apostol-type polynomials (TEFATP), denoted by which are defined by
We can obtain some definitions and some results for the TEFATP, these results are listed in Table 5.
4.4. Generalized Hermite Fubini Apostol-Type Polynomials
Setting in (22), gives generalized Hermite Fubini Apostol-type polynomials (GHFATP), denoted by which are defined by
4.5. Hermite-Appell Fubini Apostol Type Polynomials
Setting in (22), gives Hermite-Appell Fubini Apostol-type polynomials (HAFATP), denoted by which are defined by
We can obtain some definitions and some results for the TEFATP, these results are listed in Table 6.
4.6. Generalized Legendre-fubini Based Apostol Type Polynomials
Setting and in (19), gives Generalized Legendre-Fubini based Apostol-type polynomials (GLFATP), denoted by which are defined by
Example 1. If and , we obtain generalized Legendre-Based Apostol-type polynomials, see [31].
5. Probabilistic Application
5.1. Degenerate Bell Distribution (DBE)
Definition 4. Let X be a random variable which follows DBE distribution with parameters and . It is denoted by . Then its probability mass function (pmf) iswhere , is degenerate Bell polynomial in (14) and is exponential functions in (7)
Submodels:1If , we obtain Bell distribution, see [16].2If , we obtain Bell–Touchard distribution, see [17].
5.1.1. Some Structural Properties of DBE Distribution
Some statistical properties were presented through the following theorems and remarks.
Theorem 10. Let where , then, the probability generating function is
Proof. Fromand using the expansion of degenerate Bell polynomials, we obtain (58).
Corollary 4. The mean and variance of can be derived from . We have
Remark 3. The over dispersion index of DBE is given byIt follows that for . Then, DBE may be suitable for modelling count data with over dispersion.
Theorem 11. Let be independent DBE distribution and . Then, pmf of X iswhere and are positive parameters and is generalized Apostol-type numbers.
Proof. From (49), we obtainBy comparing coefficients of on both sides, we obtain (63).
Theorem 12. If is sum of zero truncated poisson random variables with parameter , see [32] and is generalized Hermite random variable with two parameter . Then, is a random variable with parameters , has probability mass function is defined as
6. Conclusions
In this paper, we introduced and investigated a new class of generalized Fubini-type polynomials. We proved some relations, identities, and summation formulae for these polynomials. Special families of polynomials are obtained for the general class of polynomials. Also, we derived explicit formulae for polynomials. Finally, probabilistic applications and some probabilistic properties are obtained.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest with this study.
Acknowledgments
The author would like to thank the Editor-in-Chief, and the anonymous referees for their careful reading and constructive comments and suggestions, which greatly improved the presentation of the paper.