- Open Access
Generating function for q-Eulerian polynomials and their decomposition and applications
Fixed Point Theory and Applications volume 2013, Article number: 72 (2013)
The aim of this paper is to define a generating function for q-Eulerian polynomials and numbers attached to any character χ of the finite cyclic group G. We derive many functional equations, q-difference equations and partial deferential equations related to these generating functions. By using these equations, we find many properties of q-Eulerian polynomials and numbers. Using the generating element of the finite cyclic group G and the generating element of the subgroups of G, we show that the generating function with a conductor f can be written as a sum of the generating function with conductors which are less than f.
MSC:05A40, 11B83, 11B68, 11S80.
The theory of the family of Eulerian polynomials and numbers (Frobenius-Euler polynomials and numbers) has become a very important part of analytic number theory and other sciences, for example, engineering, computer, geometric design and mathematical physics. Euler numbers are related to the Brouwer fixed point theorem and vector fields. Therefore, q-Eulerian type numbers may be related to Brouwer fixed point theorem and vector fields .
Recently, many different special functions have been used to define and investigate q-Eulerian numbers and polynomials, see details [1–39]. Therefore, we construct and investigate various properties of q-Eulerian numbers and polynomials, which are related to the many known polynomials and numbers such as Frobenius-Euler polynomials and numbers, Apostol-Euler polynomials and numbers, Euler polynomials and numbers.
Recently in [2–4], the authors defined a relationship between algebra and analysis. In detail, they made a new approximation between the subgroup and monoid presentation and special generating functions (such as Stirling numbers, Array polynomials etc.). In this paper, since our priority aim is to define special numbers and polynomials, it is worth depicting these references as well. Then in this paper, applying any group character χ of the finite cyclic group G to a special generating function (which has been defined in ), we give a generalization of q-Eulerian polynomials and numbers (q-Apostol-type Frobenius-Euler polynomials and numbers) and investigate their properties and some useful functional equations. Using a generating element of the subgroups of G and a generating element of G, we also decompose our generating function attached to the characters of G, and so we obtain a new decomposition of q-Eulerian numbers and polynomials attached to the characters of a subgroup of G and the Dirichlet character of G. This decomposition enables us to compute q-Apostol-type Frobenius-Euler polynomials and numbers more easily.
Throughout this paper, we assume that , the set of complex numbers, with
We use the following standard notions.
, and also, as usual, ℝ denotes the set of real numbers, denotes the set of positive real numbers and ℂ denotes the set of complex numbers.
1.1 Characters of a group G
We recall the definition and some properties of the character of an arbitrary group (see ).
A non-zero complex-valued function χ defined on G is called a character of G if for all ,
In particular, if G is a finite group with the identity element e, then and is a root of unity. In [, Theorem 6.8], it is proved that a finite abelian group G of order n has exactly n distinct characters.
Let G be the group of reduced residue classes module positive integer f. Corresponding to each character χ of G, the Dirichlet character is defined as follows:
Hence it is easily observed that
for all . In this note, f is called the conductor of the character of a group G of reduced residue classes module a positive integer f.
1.2 q-Eulerian polynomials and numbers
In , Simsek defined and studied some properties of q-Apostol type Frobenius-Euler polynomials and numbers.
Definition 1.1 Let () and .
The q-Apostol type Frobenius-Euler numbers
are defined by means of the following generating function:
The q-Apostol type Frobenius-Euler polynomials
are defined by means of the following generating function:(1)
2 q-Eulerian polynomials and numbers attached to any character
In this section, we define q-Eulerian polynomials and numbers attached to any character of the finite cyclic group G. Our new generating functions are related to nonnegative real parameters.
Definition 2.1 Let ( and ), , (). Let χ be a character of a finite cyclic group G with the conductor f.
The q-Eulerian numbers attached to the character χ
are defined by means of the following generating function:(2)
The q-Eulerian polynomials attached to the character χ
are defined by means of the following generating function:(3)
It is observed that
Upon setting in (3), we compute a q-Eulerian number attached to the character χ as follows:
By using the conductor f of the character χ and combining with , we modify Equation (2) and Equation (3), respectively, as follows:
Therefore, we provide the following relationships between q-Eulerian numbers and q-Eulerian numbers attached to the character χ.
Theorem 2.2 Let . Then we have
Proof By using (4), we deduce that
which, by comparing the coefficient on the both sides of the above equations, yields the first assertion of Theorem 2.2.
The second assertion (ii) is proved with the same argument. □
By Theorem 2.2, we also compute a q-Eulerian number attached to the character χ as follows:
Now, we turn our attention to the following generation function defined in  since we need this generating function frequently to give some functional equations for a q-Eulerian number and polynomials attached to the character χ.
Let . The number is defined by means of the following generating function:
The polynomials are defined by means of the following generating function:
Since we need this generating function frequently in this paper, we use the notation
and so it follows that .
By using the following well-known identity:
in (3), we verify the following functional equation:
Hence we have the following theorem.
Theorem 2.3 Let . Then we have
Proof By applying the Cauchy product to (7), we deduce that
Let . Then it follows that
By comparing the coefficient of on both sides of the above equation, we obtain our desired result. □
Upon setting in (7), we get the following functional equation:
By substituting in Theorem 2.3, the following theorem is easily proved.
Theorem 2.4 Let . Then we have
So that we obtain a q-difference equation for q-Eulerian polynomials attached to the character χ, we study the following equations:
which, in light of the Cauchy product of the three series , and
yield the following theorem.
Theorem 2.5 Let . Then we have
Now, we turn our attention to studying the derivative of the polynomials
and substituting in (8), we get that
Hence, using the induction method, we arrive at the following result.
Theorem 2.6 Let . Then we have
Now we give a generalization of the Raabe formula by the following theorem.
Theorem 2.7 Let . Then we have
We start the proof with defining the character
with the conductor . On the other hand, we derive that
Then by using (6), it follows that
Now, we are ready to prove our result.
Hence, comparing the coefficient of on both sides yields the assertion of this theorem. □
3 Decomposition of the generating function
In this section, using the generating element of the finite cyclic group G and the generating element of the subgroups of G, we show that the generating function with a conductor f can be written as a sum of the generating function with conductors which are less than f.
In this section, we use the following notations otherwise stated:
denotes the integer such that for all ,
We start to recall the fact that
for positive integers x and y. In particular, we get that
whenever x and y are distinct prime numbers.
Theorem 3.1 With the above notations, we get
Proof We start to recall the fact that for sets A, B,
Then by using De-Morgan’s law of sets, we get the following equality:
Now we use induction on n.
If , then and . We have
We construct the following sets:
By using De-Morgan’s law of sets, we verify the following equalities:
and for all , it is clear that .
Now assume that it is true for the set with elements. Hence, the hypothesis for the sets and is true, and we get that
Hence it follows
Example 3.2 Let , , be distinct prime integers. Then we have
Lemma 3.3 Let be a prime number for all i and
Then and is an empty set.
Proof By using the fact if and only if if and only if for all for , we get that
Let . Then we get that for all i and so .
If , then there is such that is . Therefore
Let . Then there is such that . This means that and so , a contradiction. Thus the proof is completed. □
By using the Lemma 3.3, we have one of the main results in this section.
Theorem 3.4 With the above notations, we get that
As stated before, to decompose the generating function of q-Eulerian polynomials attached to the character χ, now we need to compute the following relation:
Also, we define the character with the conductor h such as
where and . Then
Hence, we combine the above equation with the generating function
Now we are ready to state the main result without the proof in this section.
Theorem 3.5 Let , where is a prime integer. Then
where is a Dirichlet character corresponding to the character χ.
Example 3.6 Let , where is a prime integer. Then
Simsek Y: Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications. Fixed Point Theory Appl. 2013., 2013: Article ID 87. doi:10.1186/1687–1812–2013–87
Cangül IN, Cevik AS, Simsek Y: Analysis approach to finite monoids. Fixed Point Theory Appl. 2013., 2013: Article ID 15. doi:10.1186/1687–1812–2013–15
Cevik AS, Cangül IN, Simsek Y: A new approach to connect algebra with analysis: relationship and applications between presentations and generating functions. Bound. Value Probl. 2013., 2013: Article ID 51. doi:10.1186/1687–2770–2013–51
Cevik AS, Das KC, Simsek Y, Cangül IN: Some array polynomials over special monoids presentations. Fixed Point Theory Appl. 2013., 2013: Article ID 44. doi:10.1186/1687–1812–2013–44
Simsek Y: Generating functions for q -Apostol type Frobenius-Euler numbers and polynomials. Axioms 2012, 1: 395–403. doi:10.3390/axioms1030395 10.3390/axioms1030395
Apostol TM: Introduction to Analytic Number Theory. Springer, New York; 1976.
Luo QM, Srivastava HM: Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials. Comput. Math. Appl. 2005, 10: 631–642.
Luo QM, Srivastava HM: Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. J. Math. Anal. Appl. 2005, 308: 290–302. 10.1016/j.jmaa.2005.01.020
Luo QM: Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions. Taiwan. J. Math. 2006, 10(4):917–925.
Luo QM, Srivastava HM: Some relationships between Apostol-Bernoulli and Apostol-Euler polynomials. Comput. Math. Appl. 2006, 51: 631–642. 10.1016/j.camwa.2005.04.018
Luo QM, Srivastava HM: q -extensions of some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials. Taiwan. J. Math. 2011, 15: 241–257.
Luo QM: Some results for the q -Bernoulli and q -Euler polynomials. J. Math. Anal. Appl. 2010, 363: 7–18. 10.1016/j.jmaa.2009.07.042
Luo QM: An explicit relationships between the generalized Apostol-Bernoulli and Apostol-Euler polynomials associated with λ -Stirling numbers of the second kind. Houst. J. Math. 2010, 36: 1159–1171.
Luo QM: Some formulas for the Apostol-Euler polynomials associated with Hurwitz zeta function at rational arguments. Appl. Anal. Discrete Math. 2009, 3: 336–346. 10.2298/AADM0902336L
Choi J, Anderson PJ, Srivastava HM: Carlitz’s q -Bernoulli and q -Euler numbers and polynomials and a class of generalized q -Hurwitz zeta functions. Appl. Math. Comput. 2009, 215: 1185–1208. 10.1016/j.amc.2009.06.060
Kim T, Jang CL, Park HK: A note on q -Euler and Genocchi numbers. Proc. Jpn. Acad. 2001, 77: 139–141. 10.3792/pjaa.77.139
Simsek Y, Bayad A, Lokesha V: q -Bernstein polynomials related to q -Frobenius-Euler polynomials, l -functions, and q -Stirling numbers. Math. Methods Appl. Sci. 2012, 35: 877–884. 10.1002/mma.1580
Srivastava HM, Choi J: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam; 2012.
Srivastava HM: Some generalizations and basic (or q -) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci. 2011, 5: 390–444.
Kim T: A note on some formulae for the q -Euler numbers and polynomials. Proc. Jangjeon Math. Soc. 2006, 9: 227–232. arXiv:math/0608649
Kim T: The modified q -Euler numbers and polynomials. Adv. Stud. Contemp. Math. 2008, 16: 161–170. arXiv:math/0702523
Kim T: A note on the alternating sums of powers of consecutive q -integers. Adv. Stud. Contemp. Math. 2005, 11: 137–140. arXiv:math/0604227v1 [math.NT]
Carlitz L: q -Bernoulli and Eulerian numbers. Trans. Am. Math. Soc. 1954, 76: 332–350.
Choi J, Anderson PJ, Srivastava HM: Some q -extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n , and the multiple Hurwitz zeta function. Appl. Math. Comput. 2008, 199: 723–737. 10.1016/j.amc.2007.10.033
Kim T: On a q -analogue of the p -adic log gamma functions and related integrals. J. Number Theory 1999, 76: 320–329. 10.1006/jnth.1999.2373
Jang LC, Kim T: q -analogue of Euler-Barnes’ numbers and polynomials. Bull. Korean Math. Soc. 2005, 42: 491–499.
Kim T: On the q -extension of Euler and Genocchi numbers. J. Math. Anal. Appl. 2007, 326: 1458–1465. 10.1016/j.jmaa.2006.03.037
Kim T, Jang LC, Rim SH, Pak HK: On the twisted q -zeta functions and q -Bernoulli polynomials. Far East J. Appl. Math. 2003, 13: 13–21.
Kim T, Rim SH: A new Changhee q -Euler numbers and polynomials associated with p -adic q -integral. Comput. Math. Appl. 2007, 54: 484–489. 10.1016/j.camwa.2006.12.028
Ozden H, Simsek Y: A new extension of q -Euler numbers and polynomials related to their interpolation functions. Appl. Math. Lett. 2008, 21: 934–939. 10.1016/j.aml.2007.10.005
Ozden H, Simsek Y, Srivastava HM: A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl. 2010, 60: 2779–2787. 10.1016/j.camwa.2010.09.031
Schempp W: Euler-Frobenius polynomials. Internat. Schriftenreihe Numer. Math. 67. Numer. Methods Approx. Theory, vol. 7 (Oberwolfach, 1983) 1984, 131–138.
Shiratani K: On Euler numbers. Mem. Fac. Sci. Kyushu Univ. 1975, 27: 1–5.
Simsek Y: Twisted -Bernoulli numbers and polynomials related to twisted -zeta function and L -function. J. Math. Anal. Appl. 2006, 324: 790–804. 10.1016/j.jmaa.2005.12.057
Simsek Y: q -analogue of the twisted l -series and q -twisted Euler numbers. J. Number Theory 2005, 110: 267–278. 10.1016/j.jnt.2004.07.003
Simsek Y: On p -adic twisted q - L -functions related to generalized twisted Bernoulli numbers. Russ. J. Math. Phys. 2006, 13: 327–339.
Simsek Y, Kim T, Park DW, Ro YS, Jang LC, Rim SH: An explicit formula for the multiple Frobenius-Euler numbers and polynomials. JP J. Algebra Number Theory Appl. 2004, 4: 519–529.
Srivastava HM, Kim T, Simsek Y: q -Bernoulli numbers and polynomials associated with multiple q -zeta functions and basic L -series. Russ. J. Math. Phys. 2005, 12: 241–268.
Tsumura H: A note on q -analogues of the Dirichlet series and q -Bernoulli numbers. J. Number Theory 1991, 39: 251–256. 10.1016/0022-314X(91)90048-G
The authors declare that they have no competing interests.
Both authors completed the paper together. Both authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Alkan, M., Simsek, Y. Generating function for q-Eulerian polynomials and their decomposition and applications. Fixed Point Theory Appl 2013, 72 (2013). https://doi.org/10.1186/1687-1812-2013-72
- Euler numbers
- Frobenius-Euler numbers
- Frobenius-Euler polynomials
- q-Frobenius-Euler polynomials
- generating function
- character χ of the finite abelian groups G