Abstract. Inspired by the de…nition of degenerate multi-poly-Genocchi **polynomials** given by using the degenerate multi-polyexponential functions. In this paper, we consider a class of new generating function for the degenerate multi-poly-**Bernoulli** **polynomials**, called the type 2 degenerate multi-poly-**Bernoulli** poly- nomials by means of the degenerate multiple polyexponential functions. Then, we investigate their some properties and relations. We show that the type 2 degenerate multi-poly-**Bernoulli** **polynomials** equals a linear combination of the weighted degenerate **Bernoulli** **polynomials** and Stirling **numbers** of the …rst kind. Moreover, we provide an addition formula and a derivative formula. Furthermore, in a special case, we acquire a correlation between the type 2 degenerate multi-poly-**Bernoulli** **numbers** and degenerate Whitney **numbers**.

This paper performs a further investigation on the q -**Bernoulli** **numbers** and q - **Bernoulli** **polynomials** given by Acikgöz et al. (Adv Differ Equ, Article ID 951764, 9, 2010), some incorrect properties are revised. It is point out that the generating function for the q -**Bernoulli** **numbers** and **polynomials** is unreasonable. By using the theorem of Kim (Kyushu J Math 48, 73-86, 1994) (see Equation 9), some new generating functions for the q -**Bernoulli** **numbers** and **polynomials** are shown. Mathematics Subject Classification (2000) 11B68, 11S40, 11S80

power sums and proving that **Bernoulli** **polynomials** of odd order vanish for arguments equal to 0, 1/2, 1, we obtain easily the Faulhaber formula for pow- ers sums in term of **polynomials** in λ having coefficients depending on Z . These coefficients are found to be derivatives of odd powers sums on integers expressed in Z . By the way we obtain the link between Faulhaber formulae for powers sums on integers and on arithmetic progressions. To complete the work we propose tables for calculating in easiest manners possibly the Ber- noulli **numbers**, the **Bernoulli** **polynomials**, the powers sums and the Faulha- ber formula for powers sums.

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We present a new generating function related to the q-**Bernoulli** **numbers** and q-**Bernoulli** **polynomials**. We give a new construction of these **numbers** and **polynomials** related to the second-kind Stirling **numbers** and q-Bernstein **polynomials**. We also consider the generalized q- **Bernoulli** **polynomials** attached to Dirichlet’s character χ and have their generating function. We obtain distribution relations for the q-**Bernoulli** **polynomials** and have some identities involving q-**Bernoulli** **numbers** and **polynomials** related to the second kind Stirling **numbers** and q-Bernstein **polynomials**. Finally, we derive the q-extensions of zeta functions from the Mellin transformation of this generating function which interpolates the q-**Bernoulli** **polynomials** at negative integers and is associated with q-Bernstein **polynomials**.

**numbers** and q-**Bernoulli** **polynomials** appeared, diﬀerent properties of the q-**Bernoulli** **numbers** and q-**Bernoulli** **polynomials** have been well studied by many authors, see [] for a good introduction. In fact, the Carlitz’s q-**Bernoulli** **numbers** and q-**Bernoulli** **polynomials** can be deﬁned by the following generating functions (see [, ])

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In Section 2, we consider the generalized Carlitz q-**Bernoulli** **polynomials** in the p-adic case by means of a method provided by Kim. We obtain the generating functions of the generalized Carlitz q-**Bernoulli** **polynomials**. We shall provide some basic formulas for the generalized Carlitz q-**Bernoulli** **polynomials** which will be used to prove the main results of this article.

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As a degenerate version of Bernstein **polynomials**, the degenerate Bernstein polynomi- als were introduced recently (see (1.9)). Here we will study for the degenerate Bernstein **polynomials** some fundamental properties and identities associated with special **numbers** and **polynomials** including degenerate **Bernoulli** **polynomials** and central factorial num- bers of the second kind. Also, in the last section we will consider a matrix representation for those **polynomials**. For some recent works related to the present paper, the reader may want to see [14, 20, 22, 25, 27, 29]. The rest of this section is devoted to reviewing what we need in the following sections.

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2. Definitions of **Bernoulli** and Euler **numbers** and **polynomials** In this section, we will restate definitions of (generalized) **Bernoulli** **numbers**, (generalized) **Bernoulli** **polynomials**, (generalized) Euler **numbers**, and (generalized) Euler **polynomials** as follows. For more details, please see [1, 2, 3, 4, 10].

For a long time, **Bernoulli**, Euler and Genocchi **polynomials** and **numbers** and its generalizations have been extensively studied and investigated by many mathematicians and physicists (see [1-7]). Acikgoz et al. [1] considered the evaluation of the sum of more general series by Bernstein **polynomials**. Araci et al. [2] gave several novel identities for product of two Genocchi **polynomials** including Euler **polynomials** and **Bernoulli** **polynomials** and derived some applications for Genocchi **polynomials** to study a matrix formulation. Cheon [3] obtained a simple property of the **Bernoulli** **polynomials** and the Euler **polynomials** and also the relationship between two **polynomials** is derived. El-Makkawy et al. [5] provided a new approach in order to derive relations and identities for **Bernoulli**, Euler and Genocchi **polynomials** and **numbers** by means of the series manupulation prosedure. Kim [6] developed a formula for alternating sums of powers of consequtive integers by means of the Euler **numbers**. Shen [7] gave a formula for the sums of powers of consequtive integers by means of the **Bernoulli** **numbers**.

The manuscript of this paper as follows: In section 2, we consider generating functions for q-Hermite-Fubini **numbers** and **polynomials** and give some properties of these **numbers** and **polynomials**. In section 3, we derive summation formulas of q- Hermite-Fubini **numbers** and **polynomials** and some relationships between q-**Bernoulli** **polynomials**, q-Euler **polynomials** and q-Genocchi **polynomials** and Stirling **numbers** of the second kind.

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When it comes to Genocchi **numbers**, the most common thing comes to our mind is to research the relations between Genocchi **numbers**, **Bernoulli** **numbers** [14–16] and Euler **numbers** [14, 17]. Indeed, most researches on Genocchi **numbers** concern the relations between these three kinds of **numbers** (see for example [2–4, 18, 19]). In other words, there are many literatures that provide identities on these three kinds of **numbers**. Similarly, when it comes to Genocchi **polynomials**, the most common thing is to research on the relations between Genocchi **polynomials**, **Bernoulli** **polynomials** and Euler **polynomials** (see for example [2–4, 9, 18–21]). Even though when it comes to the generalized Genocchi **numbers** and generalized Genocchi **polynomials**, it is unavoidable to research the relations as above.

The concepts of **Bernoulli** **numbers** B n , **Bernoulli** **polynomials** B n (x), and the gen- eralized **Bernoulli** **numbers** B n (a,b) are generalized to the one B n (x;a,b,c) which is called the generalized **Bernoulli** **polynomials** depending on three positive real parameters. Numerous properties of these **polynomials** and some relationships between B n , B n (x), B n (a,b), and B n (x;a,b,c) are established.

Recently several authors have studied q-**Bernoulli** **polynomials**, q-Euler **polynomials** and various general- izations of these **polynomials** [1-15]. 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. Actually, we are motivated to write this paper from Kim’s systematic works on q-umbral theory [4-7].

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During the last three decades, applications of quantum calculus based on q-**numbers** have been studied and investigated succesfully, densely and considerably (see [4; 8; 10; 15; 16; 19-21; 27; 30; 31]). In conjunction with the motivation and inspiration of these applications, with the introduction of the (p; q)-number, many mathematicians and physicists have extensively developed the theory of post quantum calculus based on (p; q)-**numbers** along the traditional lines of classical and quantum calculus. Certainly, these (p; q)-**numbers** cannot be derived only switching q by q=p in q-**numbers**. Conversely, (p; q)-**numbers** are native general- izations of q-**numbers**, since q-**numbers** may be obtained when p = 1 in the de…nition of (p; q)-**numbers** (see [10]). In recent years, Corcino [5] studied on the (p; q)-extension of the binomial coe¢ cients and also derived some properties parallel to those of the ordinary and q-binomial coe¢ cients, comprised horizontal generating function, the triangular, vertical, and the horizontal recurrence relations, and the inverse and the orthogonality relationships. Duran et al.[6] considered (p; q)-analogs of **Bernoulli** **polynomials**, Euler poly- nomials and Genocchi **polynomials** and acquired the (p; q)-analogues of known earlier formulae. Duran and Acikgoz [7] gave (p; q)-analogue of the Apostol-**Bernoulli**, Euler and Genocchi **polynomials** and derived their some properties. Gupta [10] proposed the (p; q)-variant of the Baskakov-Kantorovich operators by means of (p; q)-integrals and also analyzed some approximation properties of them. Milovanovi´c et al. [22] introduced a generalization of Beta functions under the (p; q)-calculus and committed the integral modi…cation of the generalized Bernstien **polynomials**. Sadjang [26] satis…ed some properties of the (p; q)-derivatives and the (p; q)-integrations. As an application, he presented two (p; q)-Taylor formulas for **polynomials** and derived the fundamental theorem of (p; q)-calculus.

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Carlitz [1,2] introduced q-analogues of the **Bernoulli** **numbers** and **polynomials**. From that time on these and other related subjects have been studied by various authors (see, e.g., [3-10]). Many recent studies on q-analogue of the **Bernoulli**, Euler **numbers**, and **polynomials** can be found in Choi et al. [11], Kamano [3], Kim [5,6,12], Luo [7], Satoh [9], Simsek [13,14] and Tsumura [10].

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We derive a new matrix representation for higher-order Daehee **numbers** and **polynomials**, higher-order λ -Daehee **numbers** and **polynomials**, and twisted λ -Daehee **numbers** and **polynomials** of order k. This helps us to obtain simple and short proofs of many previous results on higher-order Daehee **numbers** and **polynomials**. Moreover, we obtain recurrence relations, explicit formulas, and some new results for these **numbers** and **polynomials**. Furthermore, we investigate the relation between these **numbers** and **polynomials** and Stirling, Nörlund, and **Bernoulli** **numbers** of higher-order. Some numerical results and program are introduced using Mathcad for generating higher-order Daehee **numbers** and **polynomials**. The results of this article generalize the results derived very recently by El-Desouky and Mustafa (Appl. Math. Sci. 9(73):3593-3610, 2015).

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Abstract. The purpose of this paper is to obtain several series identities in- volving some classes of generalized Apostol-**Bernoulli** and Apostol-Euler poly- nomials introduced lately by Srivastava et al. in [16, 17] as well as the general- ized sum of integer powers, the generalized alternating sum and the analogues of the expansions of the hyperbolic tangent and the hyperbolic cotangent. The method used is that of generating functions. It can be found that many identities recently obtained are special cases of our results.

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In this section, we study on generating functions for the central factorial **numbers** T n k ( ) , on analytic func- tions. By using these functions, we derive are some functional equations. Applying these functions and equ- ations, we give some identities and relations related to the central factorial **numbers** T n k ( ) , , which are defined by means of the following generating function (cf. [3] [19]):

In this paper, we construct generating functions for higher-order Euler-type **polynomials** and **numbers**. By using the generating functions, we obtain functional equations related to a generalized partial Hecke operator and Euler-type **polynomials** and **numbers**. A special case of higher-order Euler-type **polynomials** is eigenfunctions for the generalized partial Hecke operators. Moreover, we give not only some properties, but also applications for these **polynomials** and **numbers**.

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Abstract. In this paper, we investigate the recently introduced degener- ate Bernstein **polynomials** and operators and derive some of their proper- ties. Also, we give some properties of the degenerate Euler **numbers** and **polynomials** and their connection with the degenerate Euler **polynomials**.

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