Open Access

On Some Arithmetical Properties of the Genocchi Numbers and Polynomials

Advances in Difference Equations20092008:195049

https://doi.org/10.1155/2008/195049

Received: 31 October 2008

Accepted: 25 December 2008

Published: 11 February 2009

Abstract

We investigate the properties of the Genocchi functions and the Genocchi polynomials. We obtain the Fourier transform on the Genocchi function. We have the generating function of -Genocchi polynomials. We define the Cangul-Ozden-Simsek's type twisted -Genocchi polynomials and numbers. We also have the generalized twisted -Genocchi numbers attached to the Dirichlet's character . Finally, we define zeta functions related to -Genocchi polynomials and have the generating function of the generalized -Genocchi numbers attached to .

1. Introduction

After Carlitz introduced an interesting -analogue of Frobenius-Euler numbers in [1], -Bernoulli and -Euler numbers and polynomials have been studied by several authors. Recently, many authors have an interest in the -extension of the Genocchi numbers and polynomials(cf. [25]). Kim et al. [5] defined the -Genocchi numbers and the -Genocchi polynomials. In [3], Kim derived the -analogs of the Genocchi numbers and polynomials by constructing -Euler numbers. He also gave some interesting relations between -Euler and -Genocchi numbers. The first author et al. [6] obtained the distribution relation for the Genocchi polynomials.

The main aim of this paper is to derive the Fourier transform for the Genocchi function. Recently, Kim [7] investigated the properties of the Euler functions and derived the interesting formula related to the infinite series by using the Fourier transform for the Euler function. In this paper, we investigate some arithmetical properties of the Genocchi functions and the Genocchi polynomials.

In [8], Cangul-Ozden-Simsek constructed new generating functions of the twisted -extension of twisted Euler polynomials and numbers attached to the Dirichlet character . Cangul et al. [8] also defined the twisted -extension of zeta functions, which interpolate the twisted -extension of Euler numbers at negative integers. In this paper, we define the Cangul-Ozden-Simsek type twisted -Genocchi polynomials and numbers. We have the generating function of -Genocchi polynomials. We have the generalized twisted -Genocchi numbers attached to the Dirichlet character . We define zeta functions related to -Genocchi polynomials and we have the generating function of the generalized -Genocchi numbers attached to .

Let be a fixed odd prime number. Throughout this paper , , and will, respectively, denote the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the completion of algebraic closure of . Let be the set of natural numbers and . Let be the normalized exponential valuation of with When one talks of -extension, is variously considered as an indeterminate, a complex number or a -adic number . If , one normally assumes If , then we assume . We also use the following notations:
(1.1)
For , a fixed positive integer with , set
(1.2)
where satisfies the condition . The distribution is defined by
(1.3)

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
(1.4)
The fermionic -adic -measures on are defined as
(1.5)
and the fermionic -adic invariant -integral on is defined as
(1.6)
for . For , we note that
(1.7)

where . (For details see [144].)

In this paper, we investigate arithmetical properties of the Genocchi functions and the Genocchi polynomials. In Section 2, we derive the Fourier transform on the Genocchi function. In Section 3, we define the Cangul-Ozden-Simsek type twisted -Genocchi polynomials and numbers. We have the generating function of -Genocchi polynomials. We also have the generalized twisted -Genocchi numbers attached to . In Section 4, we define zeta functions related to -Genocchi polynomials and we have the generating function of the generalized -Genocchi numbers attached to .

2. Genocchi Numbers and Functions

The Genocchi numbers are defined as
(2.1)
where we use the technique method notation by replacing by , symbolically. From this definition, we can derive the following relation:
(2.2)
From (2.2), we note that , , and . The Genocchi polynomials are defined as
(2.3)
From (2.1) and (2.3), we can derive
(2.4)
By (2.1), it is not difficult to show that the recurrence relation for the Genocchi numbers is given by
(2.5)

where is the Kronecker symbol.

From (2.4) and (2.5), we note that
(2.6)

Thus, we obtain the following lemma.

Lemma 2.1.

For , one has .

From (2.4), we can easily derive
(2.7)

By (2.7), we obtain the following proposition.

Proposition 2.2.

For , one has
(2.8)

From now on, we assume that is the Genocchi function. Let us consider the Fourier transform for the Genocchi function as follows.

For , the Fourier transform on the Genocchi function is given by
(2.9)
where
(2.10)
From (2.8) and (2.10), we note that
(2.11)
Thus, for , we have
(2.12)
From (2.4) and (2.10), we derive
(2.13)
From (2.12) and (2.13), we can derive
(2.14)
By (2.9) and (2.14), we have that and
(2.15)

Therefore, we obtain the following theorem.

Theorem 2.3.

For with , one has
(2.16)
If we take , then we have
(2.17)

By (2.17) and Lemma 2.1, we obtain the following corollary.

Corollary 2.4.

For , one has
(2.18)
From Corollary 2.4, we note that
(2.19)
Thus, we have
(2.20)

By (2.20), we obtain the following corollary.

Corollary 2.5.

For , one has
(2.21)

3. -Extension of Twisted Genocchi Numbers and Polynomials

In this section, we will define the -extensions of twisted Genocchi numbers and polynomials which are the Cangul-Ozden-Simsek type twisted -Genocchi numbers and polynomials, respectively. We will have the generating function of -Genocchi polynomials and the generalized twisted -Genocchi numbers attached to .

Let . Then, we have from the definition of the Genocchi numbers and the fermionic -adic -integral on that
(3.1)
Thus, we obtain
(3.2)
For and , we have
(3.3)
By (3.2) and (3.3), if we take , we easily see that
(3.4)
Thus, we have
(3.5)
If , then we know that
(3.6)
Thus, we get
(3.7)
We can consider the generalized Genocchi numbers as follows:
(3.8)
where . Let with . From (3.3) and (3.8), we note that
(3.9)
By (3.8) and (3.9), it is not difficult to show that
(3.10)
Thus, the distribution relations for the Genocchi numbers and the Genocchi polynomials for with are obtained as follows (cf. [6]):
(3.11)

By using the multivariate integral, we can also consider the multiple Genocchi numbers and polynomials.

Let with be indeterminate and let . Then, we note that
(3.12)
Now, we define the Cangul-Ozden-Simsek type -Genocchi polynomials as follows:
(3.13)
From (3.13), we note that
(3.14)
Let be the space of primitive -th root of unity with
(3.15)
and let be the direct limit of , that is,
(3.16)

and then is a -adic locally constant space.

For , we define the Cangul-Ozden-Simsek type twisted -Genocchi polynomials as follows:
(3.17)
By (3.17), we have
(3.18)
From the result of Cangul et al. [8], we note that
(3.19)

where is the twisted -Euler polynomials.

Let be the Dirichlet character with conductor with . Then, we consider the generalized Genocchi numbers attached to as follows:
(3.20)

where .

From (3.3) and (3.20), we note that
(3.21)
By (3.20) and (3.21), it is not difficult to show that
(3.22)

Now, we also consider the Cangul-Ozden-Simsek type twisted -Genocchi numbers attached to as follows.

For and , we have
(3.23)
From (3.23), we have
(3.24)
From the result of Cangul et al. [8], we note that
(3.25)

where are called the generalized twisted -Euler numbers attached to .

4. Zeta Functions Related to the Genocchi Polynomials

In this section, we assume that with . Let be the generating function of -Genocchi polynomials defined as follows:
(4.1)

where .

Then, we note that
(4.2)
By (4.1) and (4.2), we easily see that
(4.3)

for . Therefore, we obtain the following proposition.

Proposition 4.1.

For , one has
(4.4)

From Proposition 4.1, we can derive the Genocchi zeta function which interpolates Genocchi polynomials related to -Genocchi polynomials at negative integers.

For , we define the Hurwitz-type Genocchi zeta functions related to -Genocchi polynomials and numbers as follows.

Definition 4.2.

For , one has
(4.5)

By Proposition 4.1 and Definition 4.2, we obtain the following theorem.

Theorem 4.3.

For , one has
(4.6)
The generating function of the generalized -Genocchi numbers attached to is given by
(4.7)
where , with . Therefore, we have
(4.8)
where is a nontrivial character with conductor with . From (4.8), it follows that
(4.9)
This is equivalent to
(4.10)
For with , let be a primitive Dirichlet character with conductor . Then, we define
(4.11)

From (4.10) and (4.11), we obtain the following theorem.

Theorem 4.4.

For , one has
(4.12)

Authors’ Affiliations

(1)
Department of Mathematics, Kyungpook National University
(2)
Division of General Education-Mathematics, Kwangwoon University

References

  1. Carlitz L: -Bernoulli numbers and polynomials. Duke Mathematical Journal 1948, 15(4):987-1000. 10.1215/S0012-7094-48-01588-9MATHMathSciNetView ArticleGoogle Scholar
  2. Kim T: Sums of powers of consecutive -integers. Advanced Studies in Contemporary Mathematics 2004, 9(1):15-18.MATHMathSciNetGoogle Scholar
  3. 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.2006.03.037MATHMathSciNetView ArticleGoogle Scholar
  4. Kim T: Note on -Genocchi numbers and polynomials. Advanced Studies in Contemporary Mathematics 2008, 17(1):9-15.MATHMathSciNetGoogle Scholar
  5. Kim T, Jang L-C, Pak HK: A note on -Euler and Genocchi numbers. Proceedings of the Japan Academy. Series A 2001, 77(8):139-141. 10.3792/pjaa.77.139MATHMathSciNetView ArticleGoogle Scholar
  6. Rim S-H, Park KH, Moon EJ: On Genocchi numbers and polynomials. Abstract and Applied Analysis 2008, 2008:-7.Google Scholar
  7. Kim T: Note on the Euler numbers and polynomials. Advanced Studies in Contemporary Mathematics 2008, 17(2):131-136.MATHMathSciNetGoogle Scholar
  8. 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/s10474-008-7139-1MathSciNetView ArticleGoogle Scholar
  9. Carlitz L: -Bernoulli and Eulerian numbers. Transactions of the American Mathematical Society 1954, 76: 332-350.MATHMathSciNetGoogle Scholar
  10. Cenkci M: The -adic generalized twisted -Euler- -function and its applications. Advanced Studies in Contemporary Mathematics 2007, 15(1):37-47.MATHMathSciNetGoogle Scholar
  11. Cenkci M, Can M: Some results on -analogue of the Lerch zeta function. Advanced Studies in Contemporary Mathematics 2006, 12(2):213-223.MATHMathSciNetGoogle Scholar
  12. Cenkci M, Can M, Kurt V: -adic interpolation functions and Kummer-type congruences for -twisted and -generalized twisted Euler numbers. Advanced Studies in Contemporary Mathematics 2004, 9(2):203-216.MATHMathSciNetGoogle Scholar
  13. Kim T: On -adic - -functions and sums of powers. Discrete Mathematics 2002, 252(1–3):179-187.MATHMathSciNetView ArticleGoogle Scholar
  14. Kim T: -Volkenborn integration. Russian Journal of Mathematical Physics 2002, 9(3):288-299.MATHMathSciNetGoogle Scholar
  15. Kim T: On Euler-Barnes multiple zeta functions. Russian Journal of Mathematical Physics 2003, 10(3):261-267.MATHMathSciNetGoogle Scholar
  16. Kim T: Analytic continuation of multiple -zeta functions and their values at negative integers. Russian Journal of Mathematical Physics 2004, 11(1):71-76.MATHMathSciNetGoogle Scholar
  17. Kim T: -Riemann zeta function. International Journal of Mathematics and Mathematical Sciences 2004, 2004(12):599-605. 10.1155/S0161171204307180MATHView ArticleGoogle Scholar
  18. 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.MATHMathSciNetGoogle Scholar
  19. Kim T: A new approach to -adic - -functions. Advanced Studies in Contemporary Mathematics 2006, 12(1):61-72.MATHMathSciNetGoogle Scholar
  20. Kim T: Multiple -adic -function. Russian Journal of Mathematical Physics 2006, 13(2):151-157. 10.1134/S1061920806020038MATHMathSciNetView ArticleGoogle Scholar
  21. Kim T: -extension of the Euler formula and trigonometric functions. Russian Journal of Mathematical Physics 2007, 14(3):275-278. 10.1134/S1061920807030041MATHMathSciNetView ArticleGoogle Scholar
  22. 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.2007.07.027MATHMathSciNetView ArticleGoogle Scholar
  23. Kim T: -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russian Journal of Mathematical Physics 2008, 15(1):51-57.MATHMathSciNetView ArticleGoogle Scholar
  24. Kim T: The modified -Euler numbers and polynomials. Advanced Studies in Contemporary Mathematics 2008, 16(2):161-170.MATHMathSciNetGoogle Scholar
  25. Kim T: On the symmetries of the -Bernoulli polynomials. Abstract and Applied Analysis 2008, 2008:-7.Google Scholar
  26. Kim T: Symmetry -adic invariant integral on for Bernoulli and Euler polynomials. Journal of Difference Equations and Applications 2008, 14(12):1267-1277. 10.1080/10236190801943220MATHMathSciNetView ArticleGoogle Scholar
  27. Kim T: Euler numbers and polynomials associated with zeta functions. Abstract and Applied Analysis 2008, 2008:-11.Google Scholar
  28. Kim T: A note on some formulae for the -Euler numbers and polynomials. Proceedings of the Jangjeon Mathematical Society 2006, 9(2):227-232.MATHMathSciNetGoogle Scholar
  29. Kim T: A note on -Volkenborn integration. Proceedings of the Jangjeon Mathematical Society 2005, 8(1):13-17.MATHMathSciNetGoogle Scholar
  30. Kim T: On the Sehee integral representation associated with -Riemann zeta function. Proceedings of the Jangjeon Mathematical Society 2004, 7(2):125-127.MATHMathSciNetGoogle Scholar
  31. Kim T, Rim S-H, Simsek Y: A note on the alternating sums of powers of consecutive -integers. Advanced Studies in Contemporary Mathematics 2006, 13(2):159-164.MATHMathSciNetGoogle Scholar
  32. Kim T, Simsek Y: Analytic continuation of the multiple Daehee - -functions associated with Daehee numbers. Russian Journal of Mathematical Physics 2008, 15(1):58-65.MATHMathSciNetView ArticleGoogle Scholar
  33. Kim Y-H, Kim W, Jang L-C: On the -extension of Apostol-Euler numbers and polynomials. Abstract and Applied Analysis 2008, 2008:-10.Google Scholar
  34. Jang L-C: Multiple twisted -Euler numbers and polynomials associated with -adic -integrals. Advances in Difference Equations 2008, 2008:-11.Google Scholar
  35. Ozden H, Cangul IN, Simsek Y: Multivariate interpolation functions of higher-order -Euler numbers and their applications. Abstract and Applied Analysis 2008, 2008:-16.Google Scholar
  36. 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.2007.10.005MathSciNetView ArticleGoogle Scholar
  37. Ozden H, Simsek Y, Cangul IN: Euler polynomials associated with -adic -Euler measure. General Mathematics 2007, 15(2):24-37.MathSciNetGoogle Scholar
  38. Ozden H, Simsek Y, Rim S-H, Cangul IN: A note on -adic -Euler measure. Advanced Studies in Contemporary Mathematics 2007, 14(2):233-239.MathSciNetGoogle Scholar
  39. Ryoo CS, Song H, Agarwal RP: On the roots of the -analogue of Euler-Barnes' polynomials. Advanced Studies in Contemporary Mathematics 2004, 9(2):153-163.MATHMathSciNetGoogle Scholar
  40. Simsek Y: Theorems on twisted -function and twisted Bernoulli numbers. Advanced Studies in Contemporary Mathematics 2005, 11(2):205-218.MATHMathSciNetGoogle Scholar
  41. Simsek Y: On -adic twisted - -functions related to generalized twisted Bernoulli numbers. Russian Journal of Mathematical Physics 2006, 13(3):340-348. 10.1134/S1061920806030095MATHMathSciNetView ArticleGoogle Scholar
  42. 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.MATHMathSciNetGoogle Scholar
  43. 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.MATHMathSciNetGoogle Scholar
  44. 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.MATHMathSciNetGoogle Scholar

Copyright

© K. H. Park and Y.-H. Kim. 2008

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.