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On Interpolation Functions of the Generalized Twisted -Euler Polynomials
Journal of Inequalities and Applications volume 2009, Article number: 946569 (2009)
Abstract
The aim of this paper is to construct -adic twisted two-variable Euler-(,)--functions, which interpolate generalized twisted (,)-Euler polynomials at negative integers. In this paper, we treat twisted (,)-Euler numbers and polynomials associated with -adic invariant integral on . We will construct two-variable twisted (,)-Euler-zeta function and two-variable (,)--function in Complex -plane.
1. Introduction
Tsumura and Young treated the interpolation functions of the Bernoulli and Euler polynomials in [1, 2]. Kim and Simsek studied on -adic interpolation functions of these numbers and polynomials [3–48]. In [49], Carlitz originally constructed -Bernoulli numbers and polynomials. Many authors studied these numbers and polynomials [4, 28, 38, 41, 50]. After that, twisted -Bernoulli and Euler numbers(polynomials) were studied by several authors [1–32, 32–65]. In [62], Whashington constructed one-variable -adic--function which interpolates generalized classical Bernoulli numbers at negative integers. Fox introduced the two-variable -adi -functions [53]. Young defined -adic integral representation for the two-variable -adic -functions [64]. Furthermore, Kim constructed the two-variable -adic --function, which is interpolation function of the generalized -Bernoulli polynomials [8]. This function is the -extension of the two-variable -adic -function. Kim constructed -extension of the generalized formula for two-variable of Diamond and Ferrero and Greenberg formula for two-variable -adic -function in the terms of the -adic gamma and log-gamma functions [8]. Kim and Rim introduced twisted -Euler numbers and polynomials associated with basic twisted --functions [28]. Also, Jang et al. investigated the -adic analogue twisted --function, which interpolates generalized twisted -Euler numbers attached to Dirichlet's character [55]. Kim et al. have studied two-variable -adic -functions, which interpolate the generalized Bernoulli polynomials at negative integers. In this paper, we will construct two-variale -adic twisted Euler --functions. This functions interpolation functions of the generalized twisted -Euler polynomials.
Let be a fixed odd prime number. Throughout this paper and will respectively denote the ring of rational integers, the ring of -adic rational integers, the field of -adic rational numbers and the completion of the algebraic closure of . Let be the normalized exponential valuation of such that . If , then . If , we normally assume , so that for . Throughout this paper we use the following notations (cf. [1–32, 32–48, 50, 51, 54–65]):
Hence, , for any with in the present -adic case.
For a fixed positive integer with , set
where satisfies the condition . The distribution is defined by
We say that is uniformly differential function at a point , and we write , if the difference quotients, have a limit as .
For , the -adic invariant -integral on is defined as [4, 18]
The fermionic -adic -measures on is defined as (cf. [14–16, 18, 22, 28])
for . For , the ferminoic -adic invariant -integral on is defined as
which has a sense as we see readily that the limit is convergent. For , we note that (cf. [14, 16, 18, 22, 28])
From the fermionic invariant integral on , we derive the following integral equation (cf. [14, 35]):
where .
2. Twisted -Euler Numbers and Polynomials
In this section, we will treat some properties of twisted -Euler numbers and polynomials associated with -adic invariant integral on . From now on, we take and with . Let be the space of primitive th root of unity,
Then, we denote
Hence is a -adic locally constant space. For , we denote by defined by , the locally constant function. If we take , then we have (cf. [35])
By induction in (1.8), Kim constructed the following useful identity (cf. [14, 28]):
where . From (2.4), if is odd, then we have
If we replace by into (2.5), we obtain
Let . Let be a Dirichlet's character of conductor , which is any multiple of with . By substituting into (2.6), we have
Remark 2.1.
In complex case, the generating function of the Euler numbers is given by (cf. [28])
By using Taylor series of , then we can define the generalized twisted Euler numbers attached to as follows (cf. [55]):
In [8], -Euler numbers were defined by
where and . In particular, if we take , then . These numbers are called -Euler numbers.
By using iterative method of -adic invariant integral on in the sense of fermionic, we define twisted -Euler numbers as follows (cf. [55]):
For and , we have that (cf. [55])
where with .
Let be the generating function of in complex plane as follows (cf. [55]):
Let be the Dirichlet's character with conductor with . Then the generalized twisted -Euler polynomials attached to is given by as follows:
For ,
where is any multiple of with and .
Then the distribution relation of the generalized twisted -Euler polynomials is given by as follows (cf. [14]):
3. Two-Variable Twisted -Euler-Zeta Function and --Function
In this section, we will construct two-variable twisted -Euler-zeta function and two-variable --function in Complex -plane. We assume with .
Firstly, we consider twisted -Euler numbers and polynomials in as follows (cf. [55]):
where and is th root of unity. In particular, if we take , then we have . These numbers are called twisted Euler numbers. By using derivative operator, we have .
From (3.1), we can define Hurwitz-type twisted -Euler-zeta function as follows (cf. [55]):
where and . Note that if in (3.2), then we see that the twisted -Euler-zeta function is defined by (cf. [28, 55])
For , we know (cf. [28])
From now on, we will define the two-variable --functions which interpolates the generalized -Euler polynomials.
Definition 3.1.
Let be the Dirichlet's character with conductor with . For and , we define
By substituting and into (3.5), then using (3.2), we have
Thus, we see the function which interpolates the generalized -Euler polynomials as follows.
Theorem 3.2.
For , let be the Dirichlet's character with conductor with . Then one has
By substituting with , into (3.7), we obtain
where .
Thus, we have the following theorem.
Theorem 3.3.
For , let be the Dirichlet's character with conductor with . Then one has
Remark 3.4.
If we take in (3.5), then we have (cf. [28, 55])
From (3.9) and (3.10), we have the following corollary.
Corollary 3.5.
Let be the Dirichlet's character with conductor with . Then one has
Secondly, we will define two-variable twisted Euler --function as follows.
Definition 3.6.
Let be the Dirichlet's character with conductor with . For and with , we define
We consider the well-known identity (cf. [44, 65])
By using (3.12), we define two-variable twisted Euler --function as follows:
We will investigate the relations between and as follows.
Substituting with into (3.12), we have
Thus we obtain the following theorem.
Theorem 3.7.
For with , let be the Dirichlet character with conductor with and with . Then one has
By substituting with into (3.16) and using (3.4), we can obtain
Thus, we see that the function interpolates generalized -Euler polynomials attached to at negative integer values of as followings.
Theorem 3.8.
For , let be the Dirichlet's character with odd conductor . Then one has
Note that if we take , then Theorem 3.8 reduces to Theorem 3.3.
Let and be integers with and . For , we define partial -Hurwitz type zeta function as follows:
By substituting , we have
By substituting (3.2), for , we get
Equation (3.20) means that the function interpolates polynomials at negative integers.
From (3.16) and (3.20), we have the following theorem.
Theorem 3.9.
For with , let be the Dirichlet's character with conductor with and , is any multiple of . Then one has
Remark 3.10.
If we take in (3.22), then we have
From (2.12), if we take , then we have the following corollary.
Corollary 3.11.
For with , let be the Dirichlet's character with conductor with and , is any multiple of . Then one has
4. -Adic Twisted Two-Variable Euler -L-Functions
In [62], Washington constructed one-variable -adic--function which interpolates generalized classical Bernoulli numbers negative integers. Kim [22] investigated the -adic analogues of two-variables Euler --function. In this section, we will construct -adic twisted two-variable Euler---functions, which interpolate generalized twisted -Euler polynomials at negative integers. Our notations and methods are essentially due to Kim and Washington (cf. [22, 62]).
We assume that with , so that . Let be an odd prime number. Let denote the Teichmüller character having conductor . For an arbitrary character , we define , where , in the sense of the product of characters. Let . Then . Hence we see that
where with .
We denote the subset of by (cf. [62])
Let
be a sequence of power series, each of which converges in a fixed subset such that
(1) as and
(2)for each and , there exists such that
Then for all (cf. [2, 22, 50, 51, 60, 62]).
Let be the Dirichlet's character with conductor with and let be a positive multiple of and .
Now we set
Then is analytic for with , when . For with , we have
is analytic for . It readily follows that
is analytic for with when . Thus we see that
Let and fixed with . Then we have that
If , then , so is a multiple of . Therefore, we have
Then we note that
The difference of these equations yields
Using distribution for -Euler polynomials, we easily see that
Since , for , and , with , we have
From (4.5)–(4.14), we can derive that
Therefore we obtain the following theorem.
Theorem 4.1.
Let be a positive integral multiple of and with , and let
Then is analytic for , provides when . Furthermore, for each , we have
Thus we note that for all , where is twisted -adic Euler --function, (cf. [15, 22]).
We now generalized to two-variable -adic Euler --function, which is first defined by the interpolation function
for .
From (4.18), we have that
By using the definition of , we can express for all and with as follows:
We know that is analytic for , when . The value of is the coefficients of in the expansion of at . Using the Taylor expansion at , we see that
The -adic logarithmic function, , is the unique function that satisfies
By employing these expansion and some algebraic manipulations, we evaluate the derivative . It follows from the definition of that
Thus, we have
Since is a root of unity for , we have
Thus we have the following theorem.
Theorem 4.2.
Let be a primitive Dirichlet's character with odd conductor and let be a odd positive integral multiple of and . Then for any with , one has
References
Tsumura H: On a -adic interpolation of the generalized Euler numbers and its applications. Tokyo Journal of Mathematics 1987,10(2):281–293. 10.3836/tjm/1270134514
Young PT: Congruences for Bernoulli, Euler, and Stirling numbers. Journal of Number Theory 1999,78(2):204–227. 10.1006/jnth.1999.2401
Kim T: On a -analogue of the -adic log gamma functions and related integrals. Journal of Number Theory 1999,76(2):320–329. 10.1006/jnth.1999.2373
Kim T: -Volkenborn integration. Russian Journal of Mathematical Physics 2002,9(3):288–299.
Kim T: On Euler-Barnes multiple zeta functions. Russian Journal of Mathematical Physics 2003,10(3):261–267.
Kim T: -Riemann zeta function. International Journal of Mathematics and Mathematical Sciences 2004,2004(12):599–605. 10.1155/S0161171204307180
Kim T: Analytic continuation of multiple -zeta functions and their values at negative integers. Russian Journal of Mathematical Physics 2004,11(1):71–76.
Kim T: Power series and asymptotic series associated with the -analog of the two-variable -adic -function. Russian Journal of Mathematical Physics 2005,12(2):186–196.
Kim T: -generalized Euler numbers and polynomials. Russian Journal of Mathematical Physics 2006,13(3):293–298. 10.1134/S1061920806030058
Kim T: A new approach to -adic --functions. Advanced Studies in Contemporary Mathematics 2006,12(1):61–72.
Kim T: A note on -adic invariant integral in the rings of -adic integers. Advanced Studies in Contemporary Mathematics 2006,13(1):95–99.
Kim T: Multiple -adic -function. Russian Journal of Mathematical Physics 2006,13(2):151–157. 10.1134/S1061920806020038
Kim T: -extension of the Euler formula and trigonometric functions. Russian Journal of Mathematical Physics 2007,14(3):275–278. 10.1134/S1061920807030041
Kim T: On the analogs of Euler numbers and polynomials associated with -adic -integral on at . Journal of Mathematical Analysis and Applications 2007,331(2):779–792. 10.1016/j.jmaa.2006.09.027
Kim T: On -adic --functions and sums of powers. Journal of Mathematical Analysis and Applications 2007,329(2):1472–1481. 10.1016/j.jmaa.2006.07.071
Kim T: On the -extension of Euler and Genocchi numbers. Journal of Mathematical Analysis and Applications 2007,326(2):1458–1465. 10.1016/j.jmaa.2006.03.037
Kim T: A note on -adic -integral on associated with -Euler numbers. Advanced Studies in Contemporary Mathematics 2007,15(2):133–137.
Kim T: -Euler numbers and polynomials associated with -adic -integrals. Journal of Nonlinear Mathematical Physics 2007,14(1):15–27. 10.2991/jnmp.2007.14.1.3
Kim T: Euler numbers and polynomials associated with zeta functions. Abstract and Applied Analysis 2008, 2008:-11.
Kim T: -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russian Journal of Mathematical Physics 2008,15(1):51–57.
Kim T: Note on the Euler numbers and polynomials. Advanced Studies in Contemporary Mathematics 2008,17(2):109–115.
Kim T: On -adic interpolating function for -Euler numbers and its derivatives. Journal of Mathematical Analysis and Applications 2008,339(1):598–608. 10.1016/j.jmaa.2007.07.027
Kim T: A note on -Euler numbers and polyomials. Advanced Studies in Contemporary Mathematics 2008,16(2):161–170.
Kim T: The modified -Euler numbers and polynomials. Advanced Studies in Contemporary Mathematics 2008,16(2):161–170.
Kim T: On a -adic interpolation function for the -adic interpolation function for the -extension of the generalized Bernoulli polynomials and its derivative. Discrete Mathematics 2009,309(6):1593–1602. 10.1016/j.disc.2008.03.001
Kim T: Note on the Euler -zeta functions. Journal of Number Theory. In press Journal of Number Theory. In press
Kim T, Choi JY, Sug JY: Extended -Euler numbers and polynomials associated with fermionic -adic -integral on .Russian Journal of Mathematical Physics 2007,14(2):160–163. 10.1134/S1061920807020045
Kim T, Rim S-H: On the twisted -Euler numbers and polynomials associated with basic --functions. Journal of Mathematical Analysis and Applications 2007,336(1):738–744. 10.1016/j.jmaa.2007.03.035
Ozden H, Cangul IN, Simsek Y: Remarks on sum of products of (,)-twisted Euler polynomials and numbers. Journal of Inequalities and Applications 2008, 2008:-8.
Ozden H, Simsek Y, Cangul IN: Euler polynomials associated with -adic -Euler measure. General Mathematics 2007,15(2):24–37.
Ozden H, Simsek Y: A new extension of -Euler numbers and polynomials related to their interpolation functions. Applied Mathematics Letters 2008,21(9):934–939. 10.1016/j.aml.2007.10.005
Cangul IN, Ozden H, Simsek Y: Generating functions of the (, ) extension of twisted Euler polynomials and numbers. Acta Mathematica Hungarica 2008,120(3):281–299. 10.1007/s10474-008-7139-1
Ozden H, Cangul IN, Simsek Y: Multivariate interpolation functions of higher-order -Euler numbers and their applications. Abstract and Applied Analysis 2008, 2008:-16.
Simsek Y, Yurekli O, Kurt V: On interpolation functions of the twisted generalized Frobenius-Euler numbers. Advanced Studies in Contemporary Mathematics 2007,15(2):187–194.
Rim S-H, Kim T: A note on -Euler numbers associated with the basic -zeta function. Applied Mathematics Letters 2007,20(4):366–369. 10.1016/j.aml.2006.04.019
Simsek Y: Theorems on twisted -function and twisted Bernoulli numbers. Advanced Studies in Contemporary Mathematics 2005,11(2):205–218.
Simsek Y: -analogue of twisted -series and -twisted Euler numbers. Journal of Number Theory 2005,110(2):267–278. 10.1016/j.jnt.2004.07.003
Simsek Y: On -adic twisted --functions related to generalized twisted Bernoulli numbers. Russian Journal of Mathematical Physics 2006,13(3):340–348. 10.1134/S1061920806030095
Simsek Y: Hardy character sums related to Eisenstein series and theta functions. Advanced Studies in Contemporary Mathematics 2006,12(1):39–53.
Simsek Y: Remarks on reciprocity laws of the Dedekind and Hardy sums. Advanced Studies in Contemporary Mathematics 2006,12(2):237–246.
Simsek Y: Twisted (,)-Bernoulli numbers and polynomials related to twisted (,)-zeta function and -function. Journal of Mathematical Analysis and Applications 2006,324(2):790–804. 10.1016/j.jmaa.2005.12.057
Simsek Y: On twisted -Hurwitz zeta function and -two-variable -function. Applied Mathematics and Computation 2007,187(1):466–473. 10.1016/j.amc.2006.08.146
Simsek Y: The behavior of the twisted -adic (,)--functions at -functions at . Journal of the Korean Mathematical Society 2007,44(4):915–929. 10.4134/JKMS.2007.44.4.915
Simsek Y: Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions. Advanced Studies in Contemporary Mathematics 2008,16(2):251–278.
Simsek Y, Kim D, Rim S-H: On the two-variable Dirichlet --series. Advanced Studies in Contemporary Mathematics 2005,10(2):131–142.
Simsek Y, Mehmet A: Remarks on Dedekind eta function, theta functions and Eisenstein series under the Hecke operators. Advanced Studies in Contemporary Mathematics 2005,10(1):15–24.
Ozden H, Cangul IN, Simsek Y: On the behavior of two variable twisted -adic Euler --functions. Nonlinear Analysis. In press Nonlinear Analysis. In press
Simsek Y, Yang S: Transformation of four Titchmarsh-type infinite integrals and generalized Dedekind sums associated with Lambert series. Advanced Studies in Contemporary Mathematics 2004,9(2):195–202.
Carlitz L: -Bernoulli and Eulerian numbers. Transactions of the American Mathematical Society 1954, 76: 332–350.
Cenkci M, Can M: Some results on -analogue of the Lerch zeta function. Advanced Studies in Contemporary Mathematics 2006,12(2):213–223.
Cenkci M, Simsek Y, Kurt V: Further remarks on multiple -adic --function of two variables. Advanced Studies in Contemporary Mathematics 2007,14(1):49–68.
Dąbrowski A: A note on -adic --functions. Journal of Number Theory 1997,64(1):100–103. 10.1006/jnth.1997.2107
Fox GJ: A -adic -function of two variables. L'Enseignement Mathématique, IIe Série 2000,46(3–4):225–278.
Jang L-C, Kim S-D, Park D-W, Ro Y-S: A note on Euler number and polynomials. Journal of Inequalities and Applications 2006, 2006:-5.
Jang L-C, Kurt V, Simsek Y, Rim SH: -analogue of the -adic twisted -function. Journal of Concrete and Applicable Mathematics 2008,6(2):169–176.
Koblitz N: On Carlitz's -Bernoulli numbers. Journal of Number Theory 1982,14(3):332–339. 10.1016/0022-314X(82)90068-3
Park KH, Kim Y-H: On some arithmetical properties of the Genocchi numbers and polynomials. Advances in Difference Equations 2008, -14.
Rim S-H, Park KH, Moon EJ: On Genocchi numbers and polynomials. Abstract and Applied Analysis 2008, 2008:-7.
Schikhof WH: Ultrametric Calculus: An Introduction to p-Adic Analysis, Cambridge Studies in Advanced Mathematics. Volume 4. Cambridge University Press, Cambridge, UK; 1984:viii+306.
Shiratani K, Yamamoto S: On a -adic interpolation function for the Euler numbers and its derivatives. Memoirs of the Faculty of Science, Kyushu University. Series A 1985,39(1):113–125. 10.2206/kyushumfs.39.113
Srivastava HM, Kim T, Simsek Y: -Bernoulli numbers and polynomials associated with multiple -zeta functions and basic -series. Russian Journal of Mathematical Physics 2005,12(2):241–268.
Washington LC: Introduction to Cyclotomic Fields, Graduate Texts in Mathematics. Volume 83. 2nd edition. Springer, New York, NY, USA; 1997:xiv+487.
Woodcock CF: Special -adic analytic functions and Fourier transforms. Journal of Number Theory 1996,60(2):393–408. 10.1006/jnth.1996.0130
Young PT: On the behavior of some two-variable -adic -functions. Journal of Number Theory 2003,98(1):67–88. 10.1016/S0022-314X(02)00031-8
Zhao J: Multiple -zeta functions and multiple -polylogarithms. Ramanujan Journal 2007,14(2):189–221. 10.1007/s11139-007-9025-9
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Park, K.H. On Interpolation Functions of the Generalized Twisted -Euler Polynomials. J Inequal Appl 2009, 946569 (2009). https://doi.org/10.1155/2009/946569
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DOI: https://doi.org/10.1155/2009/946569