Generating function for q-Eulerian polynomials and their decomposition and applications
© Alkan and Simsek; licensee Springer. 2013
Received: 28 December 2012
Accepted: 6 March 2013
Published: 27 March 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.
KeywordsEuler numbers Frobenius-Euler numbers Frobenius-Euler polynomials q-Frobenius-Euler polynomials q-series generating function character χ of the finite abelian groups G
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.
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 ).
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.
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.
- (i)The q-Apostol type Frobenius-Euler numbersare defined by means of the following generating function:
- (ii)The q-Apostol type Frobenius-Euler polynomialsare defined by means of the following generating function:(1)where
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.
- (i)The q-Eulerian numbers attached to the character χare defined by means of the following generating function:(2)
- (ii)The q-Eulerian polynomials attached to the character χare defined by means of the following generating function:(3)where
Therefore, we provide the following relationships between q-Eulerian numbers and q-Eulerian numbers attached to the character χ.
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. □
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.
and so it follows that .
Hence we have the following theorem.
By comparing the coefficient of on both sides of the above equation, we obtain our desired result. □
By substituting in Theorem 2.3, the following theorem is easily proved.
yield the following theorem.
Hence, using the induction method, we arrive at the following result.
Now we give a generalization of the Raabe formula by the following theorem.
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.
denotes the integer such that for all ,
whenever x and y are distinct prime numbers.
Now we use induction on n.
and for all , it is clear that .
Then and is an empty set.
Let . Then we get that for all i and so .
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.
Now we are ready to state the main result without the proof in this section.
where is a Dirichlet character corresponding to the character χ.
- 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/axioms1030395View Article
- 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.MathSciNet
- 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.2005.01.020MathSciNetView Article
- 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.2005.04.018MathSciNetView Article
- Luo QM, Srivastava HM: q -extensions of some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials. Taiwan. J. Math. 2011, 15: 241–257.MathSciNet
- Luo QM: Some results for the q -Bernoulli and q -Euler polynomials. J. Math. Anal. Appl. 2010, 363: 7–18. 10.1016/j.jmaa.2009.07.042MathSciNetView Article
- 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/AADM0902336LMathSciNetView Article
- 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.2009.06.060MathSciNetView Article
- Kim T, Jang CL, Park HK: A note on q -Euler and Genocchi numbers. Proc. Jpn. Acad. 2001, 77: 139–141. 10.3792/pjaa.77.139View Article
- 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.1580MathSciNetView Article
- 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.MathSciNet
- Kim T: A note on some formulae for the q -Euler numbers and polynomials. Proc. Jangjeon Math. Soc. 2006, 9: 227–232. arXiv:math/0608649MathSciNet
- 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.2007.10.033MathSciNetView Article
- 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.1999.2373MathSciNetView Article
- Jang LC, Kim T: q -analogue of Euler-Barnes’ numbers and polynomials. Bull. Korean Math. Soc. 2005, 42: 491–499.MathSciNetView Article
- Kim T: On the q -extension of Euler and Genocchi numbers. J. Math. Anal. Appl. 2007, 326: 1458–1465. 10.1016/j.jmaa.2006.03.037MathSciNetView Article
- 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.MathSciNet
- 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.2006.12.028MathSciNetView Article
- 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.2007.10.005MathSciNetView Article
- 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.2010.09.031MathSciNetView Article
- 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.MathSciNet
- 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.2005.12.057MathSciNetView Article
- 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.2004.07.003MathSciNetView Article
- Simsek Y: On p -adic twisted q - L -functions related to generalized twisted Bernoulli numbers. Russ. J. Math. Phys. 2006, 13: 327–339.MathSciNetView Article
- 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.MathSciNet
- 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.MathSciNet
- 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-GMathSciNetView Article
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.