Throughout this paper, we always make use of the following notation: denotes the set of natural numbers, denotes the set of nonnegative integers, denotes the set of real numbers, and denotes the set of complex numbers.
The -shifted factorial is defined by
The -numbers and -numbers factorial is defined by
respectively. The -polynomial coefficient is defined by
The -analogue of the function is defined by
In the standard approach to the -calculus two exponential function are used:
From this form we easily see that . Moreover,
where is defined by
The previous -standard notation can be found in [1].
The following elementary properties of the -Genocchi polynomials of order are readily derived from Definition 1.2. We choose to omit the details involved.
Property 1.3. Special values of the -Genocchi polynomials of order :
Property 1.4. Summation formulas for the -Genocchi polynomials of order :
Property 1.5. Difference equations:
Property 1.6. Differential relations:
Property 1.7. Addition theorem of the argument:
Property 1.8. Recurrence relationships:
2. Explicit Relationship between the -Genocchi and the -Bernoulli Polynomials
In this section we prove an interesting relationship between the -Genocchi polynomials of order and the -Bernoulli polynomials. Here some -analogues of known results will be given. We also obtain new formulas and their some special cases in the following.
Theorem 2.1. For , the following relationship
holds true between the -Genocchi and the -Bernoulli polynomials.
Proof. Using the following identity:
we have
It remains to use Property 1.8.
Since is not symmetric with respect to and , we can prove a different form of the previously mentioned theorem. It should be stressed out that Theorems 2.1 and 2.2 coincide in the limiting case when .
Theorem 2.2. For , the following relationship
holds true between the -Genocchi and the -Bernoulli polynomials.
Proof. The proof is based on the following identity:
Next we discuss some special cases of Theorems 2.1 and 2.2. By noting that
we deduce from Theorems 2.1 and 2.2 Corollary 2.3 below.
Corollary 2.3. For , the following relationship
holds true between the -Bernoulli polynomials and -Euler polynomials.
Corollary 2.4. For , the following relationship holds true:
between the classical Genocchi polynomials and the classical Bernoulli polynomials.
Note that the formula (2.9) is new for the classical polynomials.
In terms of the -Genocchi numbers , by setting in Theorem 2.1, we obtain the following explicit relationship between the -Genocchi polynomials of order and the -Bernoulli polynomials.
Corollary 2.5. The following relationship holds true:
Corollary 2.6. For the following relationship holds true:
Corollary 2.7. For the following relationship holds true:
References
G. E. Andrews, R. Askey, and R. Roy, Special Functions, vol. 71, Cambridge University Press, Cambridge, UK, 1999.
H. M. Srivastava and Á. Pintér, βRemarks on some relationships between the Bernoulli and Euler polynomials,β Applied Mathematics Letters, vol. 17, no. 4, pp. 375β380, 2004.
T. Kim, βOn the q-extension of Euler and Genocchi numbers,β Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1458β1465, 2007.
M. Cenkci, M. Can, and V. Kurt, βq-extensions of Genocchi numbers,β Journal of the Korean Mathematical Society, vol. 43, no. 1, pp. 183β198, 2006.
Y. Simsek, I. N. Cangul, V. Kurt, and D. Kim, βq-Genocchi numbers and polynomials associated with q-Genocchi-type -functions,β Advances in Difference Equations, vol. 2008, Article ID 815750, 12 pages, 2008.
J. Choi, P. J. Anderson, and H. M. Srivastava, βSome q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order , and the multiple Hurwitz zeta function,β Applied Mathematics and Computation, vol. 199, no. 2, pp. 723β737, 2008.
J. Choi, P. J. Anderson, and H. M. Srivastava, βCarlitz's q-Bernoulli and q-Euler numbers and polynomials and a class of generalized q-Hurwitz zeta functions,β Applied Mathematics and Computation, vol. 215, no. 3, pp. 1185β1208, 2009.
Q.-M. Luo and H. M. Srivastava, βq-extensions of some relationships between the Bernoulli and Euler polynomials,β Taiwanese Journal of Mathematics, vol. 15, no. 1, pp. 241β257, 2011.
N. I. Mahmudov, A New Class of Generalized Bernoulli Polynomials and Euler Polynomials, 2012.
H. M. Srivastava and C. Vignat, βProbabilistic proofs of some relationships between the Bernoulli and Euler polynomials,β European Journal of Pure and Applied Mathematics, vol. 5, no. 2, pp. 97β107, 2012.