On a class of generalized q-Bernoulli and q-Euler polynomials

Abstract. In this article, a hybrid class of the q-Hermite based Apostol type Frobenius-Euler polynomials is introduced by means of generating function and series representation. Several important formulas and recurrence relations for these polynomials are derived via different generating function methods. Further, the 2D q-Hermite based Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials are introduced and important relations for these polynomials are also established. Finally, a new class of the 2D q-Hermite based Appell polynomials is investigated as the generalization of the above polynomials. The determinant definitions for the 2D q-Hermite based Appell and related polynomials are also explored.

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“…Several important relations and formulas for these polynomials and for their generalizations are derived in [16,18,20,5,6].…”

Section: Theorem 212 the Following Relation Between The Q-hermite Bmentioning

confidence: 99%

Some results on q-Hermite based hybrid polynomials

Riyasat

Khan

2018

Glas. Mat. Ser. III

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“…Substituting x = y = 0 in the Corollary 1, we obtain B n;q that are called n-th q-Bernoulli number of order , n-th q-Euler number of order and n-th q-Genocchi number of order , respectively (see [16], [17], [18], [19], [20] and [21]). …”

Section: Corollarymentioning

confidence: 99%

“…hold true among the q-Bernoulli polynomials, q-Euler polynomials and q-Genocchi polynomials of order , see [19], [20], [21] and [16].…”

Section: Theorem 7 (Recurrence Relationships) the Followings Hold Truementioning

confidence: 99%

“…By the same motivation, in [11], Duran and Acikgoz also introduced (p; q)-analogues of Bernoulli polynomials, Euler polynomials and Genocchi polynomials and they obtained the (p; q)-analogues of known earlier formulae. One can look at the papers [1], [2], [3], [4], [6], [13], [14], [15], [16], [17], [18], [19], [20], [21], [25], [26] for more information about the special polynomials. Now we begin with the following notations: N denotes the set of the natural numbers, N 0 denotes the set of nonnegative integers, R denotes the set of real numbers and C denotes the set of complex numbers.…”

Section: Introductionmentioning

confidence: 99%

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On (p,q)-Bernoulli, (p,q)-Euler and (p,q)-Genocchi Polynomials

Duran

Açıkgöz

Aracı

2016

j comput theor nanosci

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In the present paper, we introduce a new kind of Bernoulli, Euler and Genocchi polynomials based on the (p; q)-calculus and investigate their some properties involving addition theorems, di¤erence equations, derivative properties, recurrence relationships, and so on. We also derive (p; q)-analogues of some known formulae belong to usual Bernoulli, Euler and Genocchi polynomials. Moreover, we get (p; q)-extension of Cheon's main result in [6]. Furthermore, we discover (p; q)-analogue of the main results given earlier by Srivastava and Pintér in [26].

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“…Recently several authors have studied q-Bernoulli polynomials, q-Euler polynomials and various generalizations of these polynomials [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. In the next section, we investigate modi…ed Apostol type q-Bernoulli numbers and polynomials, and we apply these numbers and polynomials to q-umbral theory which is the systematic study of q-umbral algebra.…”

Section: Introductionmentioning

confidence: 99%

Applications of <em>q</em>-Umbral Calculus to Modied Apostol Type <em>q</em>-Bernoulli Polynomials

Açıkgöz¹,

Ates²,

Duran³

et al. 2017

Preprint

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Abstract-This article aims to identify the generating function of modi…ed Apostol type q-Bernoulli polynomials. With the aid of this generating function, some properties of modi…ed Apostol type q-Bernoulli polynomials are given. It is shown that aforementioned polynomials are q-Appell. Hence, we make use of these polynomials to have applications on q-Umbral calculus. From those applications, we derive some theorems in order to get Apostol type modi…ed q-Bernoulli polynomials as a linear combination of some known polynomials which we stated in the paper.

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On a class of generalized q-Bernoulli and q-Euler polynomials

Abstract: The main purpose of this paper is to introduce and investigate a new class of generalized q-Bernoulli and q-Euler polynomials. The q-analogues of well-known formulas are derived. A generalization of the Srivastava-Pintér addition theorem is obtained.

Cited by 27 publications

References 25 publications

Some results on q-Hermite based hybrid polynomials

Some results on q-Hermite based hybrid polynomials

On (p,q)-Bernoulli, (p,q)-Euler and (p,q)-Genocchi Polynomials

Applications of <em>q</em>-Umbral Calculus to Modied Apostol Type <em>q</em>-Bernoulli Polynomials

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On a class of generalized q-Bernoulli and q-Euler polynomials

Abstract: The main purpose of this paper is to introduce and investigate a new class of generalized q-Bernoulli and q-Euler polynomials. The q-analogues of well-known formulas are derived. A generalization of the Srivastava-Pintér addition theorem is obtained.

Cited by 27 publications

References 25 publications

Some results on q-Hermite based hybrid polynomials

Some results on q-Hermite based hybrid polynomials

On (p,q)-Bernoulli, (p,q)-Euler and (p,q)-Genocchi Polynomials

Applications of <em>q</em>-Umbral Calculus to Modi ed Apostol Type <em>q</em>-Bernoulli Polynomials

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Product

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Applications of <em>q</em>-Umbral Calculus to Modied Apostol Type <em>q</em>-Bernoulli Polynomials