# bernoulli polynomials and numbers

## Top PDF bernoulli polynomials and numbers:

### A NOTE ON TYPE 2 DEGENERATE MULTI-POLY-BERNOULLI POLYNOMIALS AND NUMBERS

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.

### q Bernoulli numbers and q Bernoulli polynomials revisited

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

### The Powers Sums, Bernoulli Numbers, Bernoulli Polynomials Rethinked

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.

### A New Approach to Bernoulli Numbers and Bernoulli Polynomials Related to Bernstein Polynomials

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.

### Symmetric identities for Carlitz’s q Bernoulli numbers and 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 [, ])

### A note on Carlitz q Bernoulli numbers and polynomials

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.

### A note on degenerate Bernstein polynomials

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.

### Relationships Between Generalized Bernoulli Numbers and Polynomials and Generalized Euler Numbers and Polynomials

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].

### SOME COMPUTATIONS BETWEEN SUMS OF POWERS OF CONSECUTIVE INTEGERS AND ALTERNATING SUMS OF POWERS OF CONSECUTIVE INTEGERS

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.

### Identities on Genocchi Polynomials and Genocchi Numbers Concerning Binomial Coefficients

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.

### Generalizations of Bernoulli numbers and polynomials

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.

### Applications of q-Umbral Calculus to Modied Apostol Type q-Bernoulli Polynomials

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].

### Unified (p, q) -analog of Apostol type polynomials of order a

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.

### Some results for the q Bernoulli, q Euler numbers and polynomials

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].

### New results on higher order Daehee and Bernoulli numbers and polynomials

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).

### 10. Some series identities for some special classes of Apostol-Bernoulli and Apostol-Euler polynomials related to generalized power and alternating sums

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.

### Special Numbers on Analytic Functions

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]):

### Higher-order Euler-type polynomials and their applications

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.

### A NOTE ON DEGENERATE BERNSTEIN AND DEGENERATE EULER POLYNOMIALS

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.