Multiplication formulas for Kubert functions

doi:10.1007/s11464-013-0348-0

Front Math Chin

2014, Vol. 9

Issue (1) : 101-109 https://doi.org/10.1007/s11464-013-0348-0

RESEARCH ARTICLE

Multiplication formulas for Kubert functions

Hailong LI¹, Jing MA²(

), Yuichi URAMATSU³

1. Department of Mathematics, Weinan Teachers’ University, Weinan 714000, China; 2. School of Mathematics, Jilin University, Changchun 130012, China; 3. Graduate School of Advanced Technology, Kinki University, Iizuka, Fukuoka 820-8555, Japan

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Abstract

The aim of this paper is two folds. First, we shall prove a general reduction theorem to the Spannenintegral of products of (generalized) Kubert functions. Second, we apply the special case of Carlitz’s theorem to the elaboration of earlier results on the mean values of the product of Dirichlet L-functions at integer arguments. Carlitz’s theorem is a generalization of a classical result of Nielsen in 1923. Regarding the reduction theorem, we shall unify both the results of Carlitz (for sums) and Mordell (for integrals), both of which are generalizations of preceding results by Frasnel, Landau, Mikolás, and Romanoff et al. These not only generalize earlier results but also cover some recent results. For example, Beck’s lamma is the same as Carlitz’s result, while some results of Maier may be deduced from those of Romanoff. To this end, we shall consider the Stiletjes integral which incorporates both sums and integrals. Now, we have an expansion of the sum of products of Bernoulli polynomials that we may apply it to elaborate on the results of afore-mentioned papers and can supplement them by related results.

Keywords Kubert function multiplication formula integral formula Bernoulli polynomial mean value

Corresponding Author(s): MA Jing,Email:jma@jlu.edu.cn

Issue Date: 01 February 2014

Cite this article:

Hailong LI,Jing MA,Yuichi URAMATSU. Multiplication formulas for Kubert functions[J]. Front Math Chin, 2014, 9(1): 101-109.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-013-0348-0
https://academic.hep.com.cn/fmc/EN/Y2014/V9/I1/101

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