1. Introduction

1.1. q-Calculus. The usual quantum calculus (or recalled q-calculus) has been extensively studied for a long time by many mathematicians, physicists and engineers. The development of q-calculus stems from the applications in many fields such as engineering, economics, math-ematics, and so on. One of the important branches of q-calculus is q-special polynomials. For example, Kim ^[18] constructed q-generalized Euler polynomials based on q-exponential function. Moreover, Srivastava et al investigated Apostol q-Bernoulli, Apostol q-Euler polynomials and Apostol q-Genocchi polynomials. This is why q-calculus is thought as one of the useful tools to study with special numbers and polynomias. For more information related these issues, see, e.g. [1,2,3,5,6,8-13,16-21].

Before starting at multiple q-calculus, we first give some basic notations about q-calculus which can be found in ^[3].

For a real number (or complex number) , q-number (quantum number) is known as

(1.1)

which is also called non-symmetrical q-number. The followings can be easily derived using (1.1):

(1.2)

(1.3)

(1.4)

(1.5)

where are real or complex numbers.

The q-binomial coefficients are defined for positive integer as

(1.6)

where

The q-derivative of a function f is given as

provided exists.

For any with

For the q-commuting variables x and y such as yx = qxy, we know that

The q-integral was defined by Jackson as follows:

provided that the series on right hand side converges absolutely.

1.2. Multiple q-calculus. All notations and all corollaries written in this part have been taken from the Book of Nalci and Pashaev ^[9].

Consider basis vector with coordinates so that the multiple q-number can be de.ned as

(1.7)

which is symmetric. Hence, we can write matrix with q-numbers elements in the following form:

Diagonal terms of this matrix are defined in the limit as

(1.8)

So, by (1.8), we see that this symmetric matrix can be shown as

The followings can be easily derived using (1.7):

where n, m are real or complex numbers.

In multiple q-calculus, multiple q-derivative with base q_i, q_j is given by

representing matrix of multiple q-derivative operators which is sym-metric: where i and

Corollary 1. For N = 1 case and we have

where Also, in the case we have the standard number and the usual derivative

Corollary 2. For N = 2 case, we have

Corollary 3. Choosing = 1 and = q gives non-symmetrical case as

Corollary 4. Taking and gives symmetrical case as

The multiple q-analogue of is the polynomial

or equivalently

where x and a is commutative, xa = ax. q-multiple Binomial coefficients and multiple q-factorial are defined by

Two types of multiple q-exponential functions are de.ned by

which satisfy the following condition for commutative x and y, xy = yx

The generalization of Jackson’s integral (called multiple q-integral) is given by

Let be formal power series. Applying multiple q-integral to the both sides of gives

where C is constant.

In the next section, we will use Nalci and Pashaev’s method in order to find a systematic study of new types of the Bernoulli polynomials, Euler polynomials and Genocchi polynomials. Also we will obtain recursive formulas for these polynomials.

2. Main Results

Recently, analogues of Bernoulli, Euler and Genocchi polynomials were studied by many mathematicians ^{[1, 2, 5, 6, 11, 12, 13, 17, 18, 19, 20, 21]}. We are now ready to give the definition of generating functions, corresponding to multiple q-calculus, of Bernoulli type, Euler type and Genocchi type polynomials.

Definition 1. Let n be positive integer, we define

where , and are called, respectively, Bernoulli-type, Euler-type and Genocchi-type polynomials.

Corollary 5. Taking q_i = q_j = 1 for indexes i and j in the case N = 1 in Definition 1, we have

where B_n(x), E_n(x) and G_n(x) are called Bernoulli polynomials, Euler polynomials and Genocchi polynomials, respectively (see ^{[4, 7, 14, 15, 19]}).

Corollary 6. As a special case of Definition 1, we have

where , and are called q-Bernoulli polynomials, q-Euler polynomials and q-Genocchi polynomials, respectively (see ^{[18, 20, 21]}).

Taking x = 0 in the above definition, we have

and from the above, we write

(2.1)

From Definition 1 and (2.1), we get the following corollary.

Corollary 7. The following functional equations hold true:

By using Definition 1 and Corollary 7, it becomes

From the rule of Cauchy product, we get

(2.2)

Comparing the coefficients of in (2.2), we have

(2.3)

From this, we can get similar identities for Euler-type and Genocchi-type polynomials. Therefore, we state the following theorem.

Theorem 1. The following identities hold true:

Now we are in a position to investigate some properties of Bernoulli-type numbers and polynomials, Euler-type numbers and polynomials and Genocchi-type numbers and polynomials as follows.

From Definition 1 and by using Cauchy product, we get

If we compute both of side and then compare coefficent of , then for n > 1, we acquire

(2.4)

From this, we can get similar identities for Euler-type numbers and Genocchi-type numbers. The following theorem is an immediate consequence of Eq. (2.4).

Theorem 2. (Recurrence Formula) For n > 1, we have

It is not diffucult to show the following equality:

(2.5)

By (2.5), we get readily the following theorem.

Theorem 3. For , the followings hold true

Proof. If we change x by x + y in , then we acquire

By computing the coefficient of both of side, then we have

The others can be proved in a like manner.

Now we consider the special cases of Theorem 3 as Corollary 8 and Corollary 9.

Corollary 8. Letting y = 1 in the Theorem 3, we then get

Corollary 9. Letting x = 0 in the Theorem 3, we then get

Theorem 4. The following expressions hold true for

Proof. By using definitions of these polynomials and numbers, one can easily obtain these relations.

Theorem 5. (Identity of Symmetry) The followings hold true for :

Proof. Setting instead of x in we then get

Compairing coefficients both of side in above equality, we have desired the result. Similar to that of this proof, it can be proved for Euler-type polynomials and Genocchi-type polynomials. So we completed this proof.

Theorem 6. (Raabe’s Formula) For , the followings hold true

Proof. By using Definition 1, then we have

Similarly, we can prove this theorem for Euler-type numbers and Genocchi-type polynomials. So we omit them. Hence, we complete the proof of this theorem.

Theorem 7. The three relations between Euler-type numbers and polynomials and Genocchi-type numbers and polynomials are given by

Proof. By Definition 1, we can easily obtain these relations. So we omit the proof.

Let us now apply the multiple q-derivative , with respect to x, on the both sides of Definition 1,

Matching the coefficients of gives us

Thus we procure the following theorem.

Theorem 8. The following identities hold true:

and

Applying k-times the operator denoted by and the limit respectively, to the Definition 1, we derive that

So we conclude the following theorem.

Theorem 9. For and , we have

and

Definition 2. Let 0 < a < b. The definite multiple q-integral has the following representation:

and

Theorem 10. The following holds true:

Proof. From Definition 2, we write that

(2.6)

where equals to

(2.7)

Combining the Eq. (2.6) with the Eq. (2.7) gives us the proof of the theorem.

Theorem 11. and Then we have

Proof. By using Theorem 1, Definition 2 and for , we have

Similarly, the identities of Euler-type polynomials and Genocchi-type polynomials can be shown. Therefore, we complete the proof of theorem.

3. Further Remarks

Here we list a few values of Bernoulli-type, Euler-type and Genocchi-type numbers as follows:

Bernoulli-type number:

Table 1.

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As a special case of Table 1, we get

Table

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As a special case of Table 1, we get

Table

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Moreover the first few Bernoulli-type numbers can be shown matrix with multiple q-numbers elements in the following form

From Definition 1 and the Table 1, we easily acquire the first few Bernoulli-type poly-nomials

Bernoulli-type polynomials

Usual Bernoulli polynomials

Moreover, the first few Bernoulli-type polynomials can be shown matrix with q-numbers elements in the following form

Euler-type Numbers and Polynomials:

We begin to compute the first few value of as follows:

Table 2.

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As a special case of Table 2, we have

Table

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As a special case of Table 2, we have

Table

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Moreover the first few Euler-type numbers can be shown matrix with q-numbers elements in the following form

From Definition 1 and the Table 2, we easily acquire the first few Euler-type polynomials

Euler-type polynomials

Usual Euler polynomials

Moreover the first few Euler-type polynomials can be shown matrix with multiple q-numbers elements in the following form

Genocchi-type Numbers and Polynomials:

We begin to compute the first few value of as follows:

Table 3.

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As a special case of Table 3, we have

Table

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As a special case of Table 3, we have

Table

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Moreover the first few Genocchi-type numbers can be shown matrix with multiple q-numbers elements in the following form

From Definition 1 and the Table 3, we easily acquire the first few Genocchi-type poly-nomials

Genocchi-type polynomials

Usual Genocchi polynomials

Moreover the first few Genocchi-type polynomials can be shown matrix with multiple q-numbers elements in the following form

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