On Gould – Hopper-Based Fully Degenerate Poly-Bernoulli Polynomials with a q-Parameter

We firstly consider the fully degenerate Gould–Hopper polynomials with a q parameter and investigate some of their properties including difference rule, inversion formula and addition formula. We then introduce the Gould–Hopper-based fully degenerate poly-Bernoulli polynomials with a q parameter and provide some of their diverse basic identities and properties including not only addition property, but also difference rule properties. By the same way of mentioned polynomials, we define the Gould–Hopper-based fully degenerate (α, q)-Stirling polynomials of the second kind, and then give many relations. Moreover, we derive multifarious correlations and identities for foregoing polynomials and numbers, including recurrence relations and implicit summation formulas.


Introduction
Special functions possess a lot of importances in numerous fields of mathematics, physics, engineering and other related disciplines covering different topics such as differential equations, mathematical analysis, functional analysis, mathematical physics, quantum mechanics and so on.Particularly, the family of special polynomials is one of the most useful, widespread and applicable family of special functions.Some of the most considerable polynomials in the theory of special polynomials are Bernoulli polynomails (see [1,2]) and the generalized Hermite-Kampé de Fériet (or Gould-Hopper) polynomials (see [3]).Recently, aforementioned polynomials and their diverse extensions have been studied and developed by lots of physicsics and mathematicians, see [1,[3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references cited therein.Araci et al. [4] considered a novel concept of the Apostol Hermite-Genocchi polynomials by using the modified Milne-Thomson's polynomials and obtained several implicit summation formulae and general symmetric identities arising from different analytical means and generating functions method.Bretti et al. [6] gave multidimensional extensions of the Bernoulli and Appell polynomials by utilizing the Hermite-Kampé de Fériet polynomials and provided the differential equations, satisfing by the corresponding 2D polynomials, acquired from exploiting the factorization method.Bayad et al. [5] considered poly-Bernoulli polynomials and numbers and proved a collection of extremely important and fundamental identities satisfied by them.Cenkci et al. [7] handled poly-Bernoulli numbers and polynomials with a q parameter and investigated several aritmetical and number theoretical properties.Dattoli et al. [9] applied the method of generating function to define novel forms of Bernoulli numbers and polynomials, which were exploited to get further classes of partial sums including generalized numerous index many variable polynomials.Khan et al. [11,12] defined the Hermite poly-Bernoulli polynomials and numbers of the second kind and the degenerate Hermite poly-Bernoulli polynomials and numbers and analyzed many of their applications in combinatorics, number theory and other fields of mathematics.Kim et al. [13][14][15] dealt with the several degenerate poly-Bernoulli polynomials and numbers.Kurt et al. [16] studied on the Hermite-Kampé de Fériet based second kind Genocchi polynomials and presented diverse relationships for them.Ozarslan [19] introduced an unified family of Hermite-based Apostol-Bernoulli, Euler and Genocchi polynomials and then attained some symmetry identities between these polynomials and the generalized sum of integer powers.Ozarslan also provided explicit closed-form formulae for this unified family and proved a finite series relation between this unification and 3d-Hermite polynomials.Pathan [20] presented a new class of generalized Hermite-Bernoulli polynomials and emerged multifarious implicit summation formulae and symmetric identities by using different analytical means appying generating functions.Pathan et al. [21] introduced a new class of generalized polynomials associated with the modified Milne-Thomson's polynomials Φ (α) n (x, v) of degree n and order α and provided some of their properties.
In this paper, the usual notations C, R, Z, N and N 0 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 all nonnegative integers, respectively.
An outline of this paper is as follows.Section 2 covers the rudiments and some basic symbols and operators.Section 3 deals with the fully degenerate Gould-Hopper polynomials with a q parameter.Section 4 mainly analyzes the Gould-Hopper-based fully degenerate poly-Bernoulli polynomials with a q parameter and provides the several properties for these polynomials.Section 5 gives the definition of the Gould-Hopper-based fully degenerate (α, q)-Stirling numbers of the second kind and provides some relations for these numbers.Finally, we derive multifarious correlations and formulas including the fully degenerate Gould-Hopper polynomials with a q parameter, the Gould-Hopper-based fully degenerate poly-Bernoulli polynomials with a q parameter and the Gould-Hopper-based fully degenerate (α, q)-Stirling numbers of the second kind.

Preliminary Informations and ∆ ω Difference Operator
The Gould-Hopper family of polynomials is defined by the exponential generating function (see [6]) where j ∈ N with j ≥ 2. In the case j = 1, the corresponding bivariate polynomials are simply expressed by the Newton binomial formula.Upon setting j = 2 in (1) gives the classical Hermite polynomials H n (x, y) and the mentioned polynomials have been used to define bivariate extensions of some special polynomails, such as Bernoulli and Euler polynomials (see [9]).
For k ∈ Z with k > 1, the k-th polylogarithm function is defined by (see [5,7,10,17]) We always assume |t| < 1 along this paper.When k = 1, Li 1 (t) = − log (1 − t).In the case k ≤ 0, Li k (t) are the rational functions: Now, let us recall some basic notations and definitions the reader should know.Definition 1 (See [8,18]).Let congomega be a non-zero complex number, the ω-falling factorial is defined by The ω-Pochhammer is defined by When ω = 1, the ω-falling factorial is the usual falling factorial and the ω-Pochhammer is the usual Pochhammer [2,22] (x) (n,1) = (x Note that the ω-falling factorial and the ω-Pochhammer are linked by the relation Definition 2 (See [8,18]).The ∆ ω difference operator is defined by Proposition 1.The following difference rule holds true: Proof.We prove the result for k = 1, the general case is obtained by induction.
Proposition 2. Let f (x) be a polynomial of degree N, then the following Taylor formula holds true: Proof.Since {x (n,ω) } ∞ n=0 forms a basis of the polynomial ring, there exist constants a 0 , . . ., a N such that Applying ∆ ω j times on f (x), we get Thus (∆ ω j f )(0) = a j j! and the proposition follows.
The following Lemma will be useful in the derivation of several results.
Note that this Lemma can be extended in the following way.

The Fully Degenerate Gould-Hopper Polynomials with a q Parameter
Let n, j ∈ Z with n 0 and j > 0 and let q, x, y ∈ R/ {0} with q = 0. We define the fully degenerate Gould-Hopper polynomials with a q parameter by the following generating function to be We now examine some special cases of the fully degenerate Gould-Hopper polynomials with a q parameter as follows.
Theorem 1.The fully degenerate Gould-Hopper polynomials with a q parameter have the following representation where • is the Gauss notation, and represents the maximum integer which does not exceed the number in the square brackets.
Proof.From the generating function of the fully degenerate Gould-Hopper polynomials with a q parameter and the transformation formula (8), we get The following difference rules hold true ∆ ω y H n,q (x, y; w) = qn (j,1) H (j) n−j,q (x, y; w) .( 12) Proof.It is not difficult to see that ∆ ω xG(x, y, t) = qtG(x, y, t).Hence, we get n,q (x, y; w) n,q (x, y; w) n−1,q (x, y; w) t n n! .
Then (11) is proved.Equation ( 12) follows in the same way.
Proof.The proof follows from the equation 1 Proposition 4. The following addition formula is valid.
Proof.The proof follows from the functional equation G( Proposition 5. Let a be a non zero complex number, then the following equations is valid ).

The Gould-Hopper Based Fully Degenerate Poly-Bernoulli Polynomials with a q Parameter
Let n, k, j ∈ Z with n 0 and k, j > 0 and let q, x, y ∈ R/ {0} with q = 0. We introduce the Gould-Hopper-based fully degenerate poly-Bernoulli polynomials with a q parameter by means of the following generating function Upon setting x = 0 = y, we then get H β n,q (ω) which are called the fully degenerate poly-Bernoulli numbers with a q parameter, see [13].
Some special cases of H B (k,j) n,q (x, y) are listed in the following remark.
Proposition 6.The following connection formula holds true.
Proof.The proof follows by applying the Cauchy product.

Proposition 7.
The following difference rules apply.
n−js,q (x; ω) which gives the desired result.

The Gould-Hopper Based Fully Degenerate (α, q)-Stirling Numbers of the Second Kind
In this part, we deal with the Gould-Hopper-based fully degenerate (α, q)-Stirling numbers of the second kind and investigate their diverse relations.Definition 3. Let n, m, j ∈ Z with n m 0 and j > 0 and let q, α, x, y ∈ R/ {0} with q = 0 and α = 0.The Gould-Hopper based fully degenerate (α, q)-Stirling numbers of the second kind are defined as follows ∞ ∑ n=0 S (α,j) 2,q (n, m : x, y; ω) Remark 3.

Some Connection Formulas
In this section, we give multifarious connection formulas including the fully degenerate Gould-Hopper polynomials with a q parameter, the Gould-Hopper-based fully degenerate poly-Bernoulli polynomials with a q parameter and the Gould-Hopper-based fully degenerate (α, q)-Stirling numbers of the second kind.
We now give the following theorem.
We provide the following theorem.
We have the following theorem.
We state the following theorem.
Theorem 7. The following relation is valid where H B j n−m,q (x, y; ω) denotes the Gould-Hopper-based degenerate Bernoulli polynomials with a q parameter defined by Proof.In view of ( 14) and ( 15), we observe n,q (x, y; ω) which gives the desired result (19).
Theorem 8. We have Proof.In view of ( 14), we have n,q (x, y; ω) which completes the proof.
Thus, the proof of this theorem is completed.