A Note on -Fubini-Appell Polynomials and Related Properties

Abstract

The present article is aimed at introducing and investigating a new class of -hybrid special polynomials, namely, -Fubini-Appell polynomials. The generating functions, series representations, and certain other significant relations and identities of this class are established. Some members of -Fubini-Appell polynomial family are investigated, and some properties of these members are obtained. Further, the class of 3-variable -Fubini-Appell polynomials is also introduced, and some formulae related to this class are obtained. In addition, the determinant representations for these classes are established.

1. Introduction

The -calculus subject has gained prominence and numerous popularity during the last three decades or so (see [1–4]). The contemporaneous interest in this subject is due to the fact that -series has popped in such diverse fields as quantum groups, statistical mechanics, and transcendental number theory. The notations and definitions related to -calculus used in this article are taken from [2] (see also [5, 6]).

The -analogues of a number and the factorial function are, respectively, specified by and

The -binomial coefficient is specified by

The -analogue of is specified as

The -derivative of a function at a point is given as

The functions are called -exponential functions and satisfy the following identities:

The Fubini polynomials (FP) [7] (also known as geometric polynomials) are defined as together with the geometric series

Recently, Duran et al. [8] introduced the -analogue of the FP , denoted by and defined by means of the generating function

For , the -Fubini polynomials (q-FP) reduce to the -Fubini numbers , that is

Further, we recall the 3-variable -Fubini polynomials (3Vq-FP) [8] which are given as

Substantial properties of Fubini numbers and polynomials and their -analogue have been studied and investigated by many researchers (see [7–9] and the references cited therein). Further, these numbers and polynomials have enormous applications in analytic number theory, physics, and the other related areas.

The class of the -special polynomials such as -Fubini polynomials, -Appell polynomials, and certain members belonging to the family of -Appell polynomials such as -Bernoulli polynomials and -Euler polynomials is an expanding field in mathematics [3, 7, 8, 10, 11].

The class of -Appell polynomial sequences was established and investigated by Al-Salam [1]. These polynomials are defined by means of the generating function where is an analytic function at and denotes the -Appell numbers.

Certain significant members belonging to -Appell polynomials class are obtained based on suitable selection for the function as (1)If , the q-AP reduce to the -Bernoulli polynomials (q-BP) (see [12, 13]), that iswhere are defined by and given by denotes the -Bernoulli numbers. (2)If , the q-AP reduce to the -Euler polynomials (q-EP) (see [13, 14]), that iswhere are defined by and given by denotes the -Euler numbers.

Also, we recall the family of the numbers denoted by and defined by

In recent years, many authors have shown their interest to introduce and study new families of -special polynomials, especially the hybrid type (see [15–17] and the references therein).

The work in this article is summarized as follows: in Section 2, the replacement technique is used to introduce the class of -Fubini-Appell polynomials by combining the polynomials, -Fubini polynomials and -Appell polynomials. In Section 3, the 3-variable -Fubini-Appell polynomials are introduced which are considered as a generalization of the -Fubini-Appell polynomials. The generating relations, series representations, and some other useful properties related to these polynomials are established. In Section 4, the determinant representations of these two classes are defined. Further, certain members belonging to these polynomial families are considered, and the corresponding results are also derived.

2. -Fubini-Appell Polynomials

The -Fubini-Appell polynomials are established by means of the generating function and series representation. To achieve this, we prove the following results:

Theorem 1. The -Fubini-Appell polynomials (-FAP) are defined by means of the following generating function:

Proof. Utilizing equation (14), based on expanding the function , then replacing the powers of , i.e., by the corresponding polynomials and thereafter summing up the terms in the left-hand side of the resulting equation, we obtain that Now, denoting the resultant -FAP in the right hand side of the above equation by and utilizing equation (11) yield the assertion in equation (23).

Remark 2. Taking , the -FAP reduce to -Fubini-Appell numbers (-FAN) . Therefore, in view of equation (23), we have

Corollary 3. Taking in equation (23), we get the following generating function of the -Fubini-Bernoulli polynomials (-FBP)

Corollary 4. Taking in equation (23), we get the following generating function of the -Fubini-Euler polynomials (-FEP)

Theorem 5. The following series representation for the -FAP holds true:

Proof. In view of equations (11) and (15) and equation (23), we have which on comparing the coefficients of yield assertion in equation (28).

Theorem 6. For , the following series representation for the -FAP holds true:

Proof. In view of equations (15), (22), and (23), we can write which on comparing the coefficients of yield assertion in equation (30).

Corollary 7. Taking in equations (28) and (30), we get the following series representations of the -FBP

Corollary 8. Taking in equations (28) and (30), we get the following series representations of the -FEP :

Theorem 9. The following formula for the -FAP holds true:

Proof. Utilizing equation (23), we have which on equating the coefficients of the like powers of yields the assertion in equation (34).

Corollary 10. Taking in equations (34), we get the formula satisfied by the -FBP as

Corollary 11. Taking in equations (34), we get the formula satisfied by the -FEP as

3. 3-Variable -Fubini-Appell Polynomials

In this section, the class of 3-variable -Fubini-Appell polynomials is established, which is a generalization of the class introduced in the previous section. The generating function, series representations, and other formulae for these polynomials are obtained.

Theorem 12. The 3-variable -Fubini-Appell polynomials (3Vq-FAP) are defined by means of the following generating function:

Proof. Utilizing equations (13) and (14) and following the same method as in the proof of Theorem 1, we can get the assertion in equation (38).

Remark 13. Setting in equation (38) gives the generating function of the 2-variable -Appell polynomials (2Vq-AP) [18], that is

Corollary 14. Taking in equation (38), we get the generating function of the 3-variable -Fubini-Bernoulli polynomials (3Vq-FBP) as

Corollary 15. Taking in equation (38), we get the generating function of the 3-variable -Fubini-Euler polynomials (3Vq-FEP) as

Theorem 16. The 3Vq-FAP are defined by the series

Proof. In view of equations (13), (15), and (38), we have which on comparing the coefficients of yield assertion in equation (42).

Corollary 17. Taking in equation (42), we get the series representation of the 33Vq-FBP as

Corollary 18. Taking in equation (42), we get the series representation of the 33Vq-FEP as

Suitably using equations (4), (6), (7), (11), and (23) in generating relation (38) and then making use of the Cauchy product rule in the resultant relations and thereafter comparing the identical powers of in both sides of the resultant expressions, we get the formulae given in the following theorem.

Theorem 19. The 3Vq-FAP satisfy the following formulae

Applying the -derivatives w.r.t. and to generating relation (38), we get the results given in the following theorem.

Theorem 20. The following identities for the 3Vq-FAP hold true:

Theorem 21. The following relation for the 3Vq-FAP holds true:

Proof. Consider the identity Now, multiplying both sides of the above identity by and using equations (6), (38), and (39), we get which on equating the coefficients of the like powers of yields the assertion in equation (48).

Now, let us recall the generating function of the 2-variable -generalized tangent polynomials (2Vq-GTP) [19] given as and denotes the -generalized tangent numbers (-GTN).

Theorem 22. The following relationships between the 3Vq-FAP and 2Vq-GTP holds true:

Proof. Utilizing equations (23), (38), and (51), we have which on comparing the coefficients of yield assertion in equation (52).

Since for , the 2-variable -generalized tangent polynomials (2Vq-GTP) reduce to 2-variable -Euler polynomials [20]. Therefore, setting in equation (52) gives the following result.

Corollary 23. The following relationships between the 3Vq-FAP and 2Vq-EP holds true:

Let us recall the generating function of the 2-variable -Euler-Bernoulli polynomials (2Vq-EBP) [16] given by

Theorem 24. The following relationships between the 3Vq-FAP and 2Vq-EBP holds true:

Proof. Utilizing equations (6), (38), and (55), we have which on comparing the coefficients of yield assertion in equation (56).

Theorem 25. The following relationships between the 3Vq-FAP and 2Vq-AP holds true:

Proof. Replacing by in generating relation (38), we have Rewriting the above equation then using equations (38) and (39), we obtain which on comparing the coefficients of yield assertion in equation (58).

4. Determinant Representations

One of the significant representations of the -special polynomials is the determinant representation due to its importance for the computational and applied purposes. In 2015, Keleshteri and Mahmudov [18] established the determinant representation of the -Appell polynomials. In the section, the determinant representations of the -FAP and the 3Vq-FAP are introduced.

Definition 26. The determinant representation for the -FAP of degree is given as

Setting and in equations (61) and (62) gives the determinant representation of the -FBP as:

Definition 27. The determinant representation for the -FBP of degree is given as

Setting and in equations (61) and (62) gives the determinant representation of the -FEP as:

Definition 28. The determinant representation for the -FEP of degree is given as

Similarly, the determinant representation of the 3Vq-FAP , 3Vq-FBP , and 3Vq-FEP are established as:

Definition 29. The determinant representation for the 3Vq-FAP of degree is given as

Definition 30. The determinant representation for the 3Vq-FBP of degree is given as

Definition 31. The determinant representation for the 3Vq-FEP of degree is given as

5. Conclusions

Recently, the Fubini polynomials and their -analogue have been studied and investigated by many researchers. Motivated by various recent studies related to these type of polynomials (see for example [8, 21, 22]), in this article, we introduced two important families of -hybrid special polynomials, namely, the -Fubini-Appell polynomials and 3-variable -Fubini-Appell polynomials. Certain properties related to these families are derived.

Further investigations along the results obtained in this article, which are associated with many recent generalizations and extensions of the -Appell polynomial family, especially, the parametric types, may be worthy of consideration in future investigations.

Data Availability

There is no data availability in this manuscript.

Conflicts of Interest

The authors declare that they have no conflicts of interest.