- Open Access
On the twisted Daehee polynomials with q-parameter
© Park; licensee Springer. 2014
Received: 16 September 2014
Accepted: 17 November 2014
Published: 2 December 2014
The n th twisted Daehee numbers with q-parameter are closely related to higher-order Bernoulli numbers and Bernoulli numbers of the second kind. In this paper, we give a p-adic integral representation of the twisted Daehee polynomials with q-parameter, and we derive some interesting properties related to the n th twisted Daehee polynomials with q-parameter.
Let p be a fixed prime number. Throughout this paper, , , and will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completions of algebraic closure of . The p-adic norm is defined .
Note that for each .
When , are called the Bernoulli numbers of order r, and in the special case, , are called the ordinary Bernoulli polynomials.
where is the cyclic group of order .
When , are called the Daehee numbers with q-parameter.
where with and .
In this paper, we give a p-adic integral representation of the twisted Daehee polynomials with q-parameter, which is called the Witt-type formula for the twisted Daehee polynomials with q-parameter. We can derive some interesting properties related to the n th twisted Daehee polynomials with q-parameter.
2 Witt-type formula for the n th twisted Daehee polynomials with q-parameter
By (2.2) and (2.3), we obtain the following theorem.
By (2.4) and (2.5), we obtain the following corollary.
By (1.6), (2.7), and (2.8), we obtain the following corollary.
where n is a nonnegative integer and . In the special case, , are called the Daehee numbers of order k with q-parameter.
By (2.13), (2.14), and (2.15), we obtain the following theorem.
In the special case , are called the twisted Daehee numbers of the second kind with q-parameter.
By (1.10), it is easy to show that . Thus, from (2.19), we have the following theorem.
By (2.20) and (2.21), we obtain the following theorem.
where n is a nonnegative integer and . In the special case, , are called the higher-order twisted Daehee numbers of the second kind with q-parameter.
From (1.10), we know that . Hence, by (2.24), we obtain the following theorem.
By (2.25) and (2.26), we obtain the following theorem.
The author is grateful for the valuable comments and suggestions of the referees. This paper was supported by the Sehan University Research Fund in 2014.
- Kim T: On q -analogye of the p -adic log gamma functions and related integral. J. Number Theory 1999, 76(2):320-329. 10.1006/jnth.1999.2373MathSciNetView ArticleGoogle Scholar
- Kim T: An invariant p -adic integral associated with Daehee numbers. Integral Transforms Spec. Funct. 2002, 13(1):65-69. 10.1080/10652460212889MathSciNetView ArticleGoogle Scholar
- Kim T: q -Volkenborn integration. Russ. J. Math. Phys. 2002, 9(3):288-299.MathSciNetGoogle Scholar
- Comtet L: Advanced Combinatorics. Reidel, Dordrecht; 1974.View ArticleGoogle Scholar
- Kim T, Kim DS, Mansour T, Rim SH, Schork M: Umbral calculus and Sheffer sequences of polynomials. J. Math. Phys. 2013., 52(8): Article ID 083504Google Scholar
- Roman S: The Umbral Calculus. Dover, New York; 2005.Google Scholar
- Dolgy DV, Kim T, Lee B, Lee SH: Some new identities on the twisted Bernoulli and Euler polynomials. J. Comput. Anal. Appl. 2013, 15(3):441-451.MathSciNetGoogle Scholar
- Jeong JH, Jin JH, Park JW, Rim SH: On the twisted weak q -Euler numbers and polynomials with weight 0. Proc. Jangjeon Math. Soc. 2013, 16(2):157-163.MathSciNetGoogle Scholar
- Kim DS, Kim T: Daehee numbers and polynomials. Appl. Math. Sci. 2013, 7(120):5969-5976.MathSciNetGoogle Scholar
- Kim YH, Hwang KW: Symmetry of power sum and twisted Bernoulli polynomials. Adv. Stud. Contemp. Math. 2009, 18(2):43-48.MathSciNetGoogle Scholar
- Luo QL: Some recursion formulae and relations for Bernoulli numbers and Euler numbers of higher order. Adv. Stud. Contemp. Math. 2005, 10(1):63-70.MathSciNetGoogle Scholar
- Ozden H, Cangul IN, Simsek Y: Remarks on q -Bernoulli numbers associated with Daehee numbers. Adv. Stud. Contemp. Math. 2009, 18(1):41-48.MathSciNetGoogle Scholar
- Simsek Y: Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions. Adv. Stud. Contemp. Math. 2008, 16(2):251-278.MathSciNetGoogle Scholar
- 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 87Google Scholar
- Simsek Y: On p -adic twisted q - L -function related to generalized twisted Bernoulli numbers. Russ. J. Math. Phys. 2006, 13(3):340-348. 10.1134/S1061920806030095MathSciNetView ArticleGoogle Scholar
- Araci S: Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus. Appl. Math. Comput. 2014, 233: 599-607.MathSciNetView ArticleGoogle Scholar
- Araci S, Acikgoz M, Sen E: On the von Staudt-Clausen’s theorem associated with q -Genocchi numbers. Appl. Math. Comput. 2014, 247: 780-785.MathSciNetView ArticleGoogle Scholar
- Araci S, Bagdasaryan A, Özel C, Srivastava HM: New symmetric identities involving q -zeta type functions. Appl. Math. Inf. Sci. 2014, 8(6):2803-2808. 10.12785/amis/080616MathSciNetView ArticleGoogle Scholar
- Park JW, Rim SH, Kim J: The twisted Daehee numbers and polynomials. Adv. Differ. Equ. 2014., 2014: Article ID 1Google Scholar
- Kim DS, Kim T, Kwon HI, Seo JJ: Daehee polynomials with q -parameter. Adv. Stud. Theor. Phys. 2014, 8(13):561-569.Google Scholar
- Kim T, Lee SH, Mansour T, Seo JJ: A note on q -Daehee polynomials and numbers. Adv. Stud. Contemp. Math. 2014, 24(2):155-160.Google Scholar
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