Abstract

In this paper, we first provide the generalized degenerate Gould-Hopper polynomials via thedegenerate exponential functions and then give various relations and formulas such as addition formula andexplicit identity. Moreover, we consider the generalized Gould-Hopper based degenerate central factorialnumbers of the second kind and present several identities and relationships. Furthermore, we introducethe generalized Gould-Hopper based fully degenerate central Bell polynomials and investigated multifariouscorrelations and formulas including summation formulas, derivation rule and correlations with the Stirlingnumbers of the first kind, the generalized Gould-Hopper based degenerate central factorial numbers of thesecond kind and the generalized degenerate Gould-Hopper polynomials. We then acquire some relations withthe degenerate Bernstein polynomials for the generalized Gould-Hopper based fully degenerate central Bellpolynomials. Finally, we consider the Gould-Hopper based fully degenerate Bernoulli, Euler and Genocchipolynomials and by utilizing these polynomials, we develop some representations for the generalized Gould-Hopper based fully degenerate central Bell polynomials.

[2010] Mathematics Subject Classification. Primary 11B73; Secondary 11B68, 33B10.

1. Motivation

Special functions and special polynomials possess a lot of significance innumerous fields of physics, mathematics, applied sciences, engineering and other related research areas including, functional analysis, differential equations, quantum mechanics, mathematical analysis, mathematical physics, and so on. Especially, the family of special polynomials is one of the most applicable, widespread and useful family of special functions. Some of the most considerable polynomials in the theory of special polynomials are the generalized Hermite-Kampé de Fériet (or Gould-Hopper) polynomials (see ¹) and the Bell polynomials (see ²).

The Bell polynomials considered by Bell ² appear as a standard mathematical tool and arise in combinatorial analysis. In recent years, the usual Bell polynomials and the familiar central Bell polynomials have been extensively investigated by several mathematicians, cf. ^{2, 3, 4, 5, 6, 7, 8, 9, 10, 11} and see also the references cited therein.

In the theory of special functions and special polynomials, the degenerate forms for polynomials and functions have been worked and developed by several mathematicians cf. ^{5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} and see also the references cited therein. For example, Carlitz ¹² considered the degenerate Euler polynomials of higher order and presented diverse properties. Carlitz ¹³ introduced the degenerateStaudt-Clausen theorem and also illustrated it for the degenerate Bernoulli numbers. Kim et al. ¹⁷ introduced the degenerate Bernstein polynomials and examined recurrence relations, their generating function, symmetric identities and various connections with the earlier polynomials. Kim et al. ⁸ considered the degenerate central Bell numbers and polynomials and provided several properties, identities, and recurrence relations. Kim et al. ⁹worked on degenerate Bell numbers and polynomials and gave diverse new formulas for those numbers and polynomials. Kim et al. ¹⁸ handled multifarious explicit formulas and recurrence relationships for the degenerate Mittag-Leffler polynomials and investigated diverse relationships between Mittag-Leffler polynomials and other known families of polynomials. Kim et al. ¹⁹ introduced the degenerate gamma function and degenerate Laplace transform and proved some interesting and novel formulas.

Throughout this paper, the familiar symbols , , , and are referred to the set of all complex numbers, the set of all real numbers, the set of all integers, the set of all natural numbers and the set of allnon-negative integers, respectively.

The rest of this paper is structured as follows: Section 2 provides the definition of the generalized degenerate Gould-Hopper polynomials by means of the degenerate exponential functions and also includesvarious relations and formulas for these polynomials. Section 3 deals with the generalized Gould-Hopperbased degenerate central factorial numbers of the second kind and covers several identities and relationships. Section 4 considers the generalized Gould-Hopper based fully degenerate central Bell polynomials andpresents multifarious correlations and formulas associated with the degenerate Bernstein polynomials andthe Gould-Hopper based fully degenerate Bernoulli, Euler and Genocchi polynomials for the mentioned Bellpolynomials. The last section of this paper examines the results derived in this paper.

2. Introduction and Preliminaries

In this section, we consider the generalized degenerate Gould-Hopper polynomials via the degenerateexponential functions. Before defining these polynomials, we provide some information that we need.

The Gould-Hopper polynomials are given by means of the following Taylorseries expansion at (see ^{1, 14, 24}):

(2.1)

with For the special case the Gould-Hopper polynomialsreduce to the representation of the Newton binomial formula. When in(2.1), we get the usual Hermite polynomials denoted by that have been utilized to generalize several specialnumbers and polynomials, for instance, Bell, Euler and Bernoulli polynomialsand numbers (see ²⁵).

Here are several basic notations and definitions in order to define the generalized degenerate Gould-Hopperpolynomials.

Forthe -falling factorial is defined by ²⁶

(2.2)

The -rising factorial is given by (see ²⁶)

(2.3)

In the case r = 1, the r-falling factorial reduces to the familiar falling factorial (see ²⁶)

andr-rising factorial becomes the usual rising factorial ^{25, 26, 27}

The Stirling numbers of the first kindare defined by means of the falling factorial as follows

(2.4)

cf. ^{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 26, 27} and see also references cited therein.

The r-falling factorial and the r-rising factorial satisfy the following relation

(2.5)

Thedifference operator is defined by (see ²⁵)

(2.6)

Proposition 1.(cf. ²⁵) The following difference rule holds true:

(2.7)

The following Lemma will be useful in the derivation of several results.

Lemma 1. (cf. ¹⁴) The following elementary series manipulations hold:

(2.8)

The degenerate exponential functionfor a real numberis given by (cf. ^{5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}):

(2.9)

It is readily seen that . From (2.2) and (2.9), we obtain the followingrelation

(2.10)

satisfying

(2.11)

We now give our definition as follows.

Definition 1. Letwithand let We define the generalized degenerate Gould-Hopper polynomialsby the following generating function to be

(2.12)

We now examine some special cases of the generalized degenerate Gould-Hopper polynomials as follows.

Remark 1.

(1)When we get the fully degenerate Gould-Hopperpolynomials denoted by (see ¹⁴).

(2)Setting and we get the fully degenerateHermite polynomials denoted by (cf. ¹⁶).

(3)When and , we have theGould-Hopper polynomials denoted by(cf. ^{1, 14, 28}).

(4)When , and , we reachthe classical Hermite polynomials denoted by (see ^{1, 16, 24, 28, 29, 30, 31, 35}).

We now give four Theorems follow from Definition 1 and the transformation formula (2.8) without proofs.

Theorem 1. The generalized degenerate Gould-Hopper polynomials satisfy the following explicitformula

(2.13)

where is the Gauss notation, andrepresents the maximum integer which does not exceed the number in thesquare brackets.

Here is the inversion formula for the generalized degenerate Gould-Hopperpolynomials .

Theorem 2.The following inversion formula holds true.

(2.14)

Theorem 3.The following addition formula is valid.

(2.15)

Theorem 4.Forthe following equation holds true

(2.16)

3. The Generalized Gould-Hopper Based Degenerate Central Factorial Numbers

In this section, we perform to analyze and investigate degenerate forms of some special polynomials andnumbers. We focus on the generalized Gould-Hopper based degenerate central factorial numbers of thesecond kind. We then derive several properties and formulas for these numbers.

For non-negative integer the central factorial numbers of the secondkind are defined by the following exponentialgenerating function

(3.1)

or by recurrence relation for a fixed non-negative integer ,

(3.2)

where the notation called as the central factorialequals to with initial condition , cf. ^{5, 7, 8, 10} and see also references citedtherein.

For non-negative integer the degenerate central factorial numbers ofthe second kind are defined by thefollowing exponential generating function

(3.3)

When approaches to , the degenerate central factorial numbersof the second kind (3.3) reduces to the central factorial numbers ofthe second kind (3.1), namely

We are now ready to define the generalized Gould-Hopper based degenerate central factorialnumbers of the second kind.

Definition 2. Let The generalized Gould-Hopper based degenerate central factorial numbers of the second kind are introduced by means of the following generating function

(3.4)

We here analyze some circumstances of the generalized Gould-Hopper baseddegenerate central factorial numbers of the second kind as follows.

Remark 2.

(1) When , we get the unified degenerate central factorialnumbers of the second kind

(3.5)

(2)When , we get an extension for the -central factorialnumbers, termed the unified degenerate -central factorial numbers of thesecond kind:

(3.6)

(3) When , we get a new polynomial which is an extension of the central factorial numbers:

(3.7)

(4) When and , we get the degeneratecentral factorial numbers of the second kind in (3.3), cf. ⁸.

(5) When , generalized Gould-Hopper baseddegenerate central factorial numbers of the second kind reduce tothe -analog of the degenerate Gould-Hopper based centralfactorial numbers of the second kind denoted by , which is also novelgeneralization of the factorial numbers of the second kind in (3.1), given by

(3.8)

(6) When and ,we attain the familiar central factorial numbers of the second kind in (3.1), cf. ^{5, 7, 8, 10}.

We now investigate some properties of the generalized Gould-Hopper baseddegenerate central factorial numbers of the second kind . Hence, wegive the following Theorem 5.

Theorem 5.Forandwe have

(3.9)

(3.10)

(3.11)

Proof. In view of Definition 2 and using (3.5), (3.6) and(3.7), we write

which implies the asserted result (3.9). The equations (3.10) and (3.11) can be derived similarly. So, theproof is completed.

Here are the differentiation rules for the generalized Gould-Hopper based degenerate central factorialnumbers of the second kind.

Theorem 6.The following relation holds true for

(3.12)

(3.13)

and

(3.14)

Proof. From (3.4), we get

which implies the claimed result (3.12). The proofs of the results in (3.13) and (3.14) can be done by the similarproof method used above.

We here give the following correlation.

Theorem 7. The following correlation

(3.15)

is valid for

Proof.By Definition 2 and (2.12), we obtain

which provides the desired result (3.15).

We give the following theorem.

Theorem 8.The following relation

(3.16)

holds true for

Proof. By Definition 3, we get

which implies the desired result (3.16).

We here give the following correlation.

Theorem 9.The following correlation

(3.17)

is valid for

Proof. By Definition 2 and the identity (2.9), we obtain

which provides the desired result (3.17).

4. Construction of Generalized Gould-Hopper Based Fully Degenerate Central BellPolynomials

In this part, we introduce the generalized Gould-Hopper based fullydegenerate central Bell polynomials and investigated multifariouscorrelations and formulas including summation formulas, derivation rule andcorrelations with the Stirling numbers of the first kind, the generalizedGould-Hopper based degenerate central factorial numbers of the second kindand the generalized degenerate Gould-Hopper polynomials.

The classical Bell polynomials (also calledexponential polynomials) and central Bell polynomials (also called central exponential polynomials)are defined by means of the following generating functions:

(4.1)

and

(4.2)

The classical Bell numbers and usual central Bell numbers are acquired by choosing in (4.1) and(4.2), that is and , which are given bythe following exponential generating function:

(4.3)

The Bell polynomials extensively studied by Bell ² appear as a standard mathematical tool and arise incombinatorial analysis. The familiar Bell polynomials and the central Bell polynomials have been intenselystudied by many mathematicians, cf. ^{2, 3, 4, 5, 6, 7, 8, 9, 10, 11} and see also the references cited therein.The large investigations of the Bell polynomials and numbers yield a motivation to improve this mathematical tool.

The central Bell polynomials and central factorial numbers of the second kind satisfy the following relation(cf. ^{5, 7, 8, 10}).

(4.4)

The degenerate classical Bell polynomials and the degenerate central Bellpolynomials are given by the following Taylor series expansion at asfollows:

(4.5)

and

(4.6)

When in (4.5) and (4.6), the mentioned polynomials (and ) reduce to the corresponding numbers

(4.7)

termed as the degenerate Bell numbers and the degenerate central Bell numbers, respectively.

Remark 3. We note that using (2.9), the degenerate classical Bell polynomials (4.5) and the degenerate central Bell polynomials (4.6) reduce theclassical Bell polynomials (4.1) and the central Bell polynomials (4.2) in the following limit cases:

(4.8)

The degenerate central Bell polynomials and the degenerate central factorialnumbers of the second kind satisfy the following relation(cf. ⁸)

(4.9)

We are now ready to define the generalized Gould-Hopper based fully degenerate central Bell polynomialsand numbers by the following Definition 3.

Definition 3. LetThe generalized Gould-Hopper based fully degenerate central Bell polynomials are defined by the following exponential generating function

(4.10)

When the generalized Gould-Hopper based fully degenerate central Bellpolynomials reduce to the corresponding numbers

termed as generalized Gould-Hopper based fully degeneratecentral Bell numbers:

(4.11)

We now analyze various special circumstances of the generalized Gould-Hopper based fully degeneratecentral Bell polynomials as follows.

Remark 4.

(1)When , the polynomials andnumbers in (4.10) and (4.11) reduce to thegeneralized Gould-Hopper based degenerate central Bell polynomials and numbers in (4.12), which are alsonew generalizations of the central Bell polynomials and numbers in (4.2) and (4.3), given by

(4.12)

and

(2) Upon setting , the polynomials and numbers in (4.10) and (4.11) reduce to the Gould Hopper based generalized degenerate central Bellpolynomials and numbers in (4.12), which are also novel extensions of thecentral Bell polynomials andnumbers in (4.2) and (4.3), shown by

(4.13)

(3)When and , weobtain the degenerate central Bell polynomials and numbers denoted by and in (4.6) and (4.7) (cf. ⁸)

(4)Setting and we attain the degenerate central Bell polynomials and numbers denoted by and , which is different from the polynomials and numbers in (4.2) and (4.3) given by Kim et al. ⁸:

(4.14)

(5) In the special case and the polynomials and numbers in (4.10) and (4.11) reduce to the Gould Hopperbased central Bell polynomials and numbers in (4.15), which are also new generalizations of thecentral Bell polynomials andnumbers in (4.2) and (4.3), given by

(4.15)

(6) When and , we arrive at the central Bell polynomials and numbers in (4.2)and (4.3) (cf. ^{5, 7, 8, 10}).

4.1.Simple Identities for

We now list a few properties of which followstraightforwardly from Definition 3. So we omit the proofs.

Theorem 10.For we have

(4.16)

Theorem 11.The following relation

(4.17)

holds true for

We now state two summation formulas for as follows.

Theorem 12. The following summation formulas

(4.18)

and

(4.19)

are valid for

We now provide a correlation as follows.

Theorem 13.The following formula

(4.20)

holds true for

We here provide an explicit formula for asfollows.

Theorem 14.The following explicit formula

(4.21)

holds true for

4.2. A Partial Derivative for

Theorem 15 includes the partial derivative of with respect to as follows.

Theorem 15.The following relation

(4.22)

holds true for

Proof.By Definition 3 and formulas (2.9) and (2.10), we get

which means the claimed result (4.22).

4.3. Relations for

Here, we perform to get several diverse relations for with some other degenerate polynomials including degenerateBernstein, Bernoulli, Genocchi and Euler polynomials.

We firstly perform to attain some relations with the degenerate Bernstein

(4.23)

Remark 5. Upon setting in (4.23), the generation functionof degenerate Bernstein polynomials reduce to the generating function offamiliar Bernstein polynomials as follows:

which is firstly given by Acikgoz and Araci in ³².

We consider that

Hence, we arrive at the following theorem.

Theorem 16. The following correlation

(4.24)

holds true.

Let

Therefore, from (3.4) and (4.23), we obtain

and by (3.5) and (4.10), similarly

Thus, we arrive at the following theorem.

Theorem 17. The following summation equality

(4.25)

is valid.

We here generalize the classical Bernoulli , Euler and Genocchi polynomialsabove via the generalized degenerate Gould-Hopper polynomials (2.12) asfollows.

Definition 4. The generalized degenerate Gould-Hopper based\ fully degenerate Bernoulli ,Euler and Genocchi polynomials are defined by the following exponentialgenerating functions:

(4.26)

(4.27)

(4.28)

for

Remark 6. As the parameters and goes to , thegeneralized degenerate Gould-Hopper based fully degenerate Bernoulli, Eulerand Genocchi polynomials reduce to the usual Bernoulli, Euler and Genocchipolynomials, cf. ^{12, 15, 21, 23, 24, 26, 27, 29, 30, 33, 34}.

When , the polynomials in (4.26), (4.27) and (4.28)reduce to the corresponding degenerate numbers, namely , and , see ^{12, 23} and the references citedtherein for further details on the mentioned numbers.

Note that the following relation holds true as has been in the usualGenocchi and Euler numbers:

The summation formulas for the polynomials , and are stated in the following theoremwithout proofs.

Theorem 18. The following formulas are valid:

We now perform to acquire some representations for bymeans of the Gould-Hopper based fully degenerate Bernoulli, Euler andGenocchi polynomials and fully degenerate central Bell polynomials.

Theorem 19. The following correlation

(4.29)

holds true.

Proof. By (4.10) and (4.26), we get

which implies the desired result (4.29).

Theorem 20. The following summation formula

is valid.

Proof. From (4.10) and (4.27), the proof can be completed by utilizing a similar proof method in Theorem19. So we omit the proof.

Theorem 21. The following relation

holds true.

Proof. Because of (4.10) and (4.28), the proof can be done by using a similar proof method in Theorem19. So we omit the proof.

5. Conclusion

In this paper, we have first defined the generalized degenerate Gould-Hopper polynomials via the degenerate exponential functions and then have given various relations and formulas such asaddition formula and explicit identity. Also, we have defined the generalized Gould-Hopper based degenerate central factorial numbers of the second kind and have presented several identities and relationships. Wehave considered the generalized Gould-Hopper based fully degenerate central Bell polynomials and have derived multifarious correlations and formulas including summation formulas, derivation rule and correlationswith the Stirling numbers of the first kind, the generalized Gould-Hopper based degenerate central factorialnumbers of the second kind and the generalized degenerate Gould-Hopper polynomials. We then have investigated some relations with the degenerate Bernstein polynomials for the generalized Gould-Hopper basedfully degenerate central Bell polynomials. Lastly, by introducing the Gould-Hopper based fully degenerate Bernoulli, Euler and Genocchi polynomials, we have proved many representations for the generalizedGould-Hopper based fully degenerate central Bell polynomials via the introduced polynomials.

References