关于一类poly-Dedekind DC和的研究 A study on a type of poly-Dedekind type DC sums 1 first-author 马元魁，女，山东青岛人，教授，从事数论及其应用研究，mayuankui@xatu.edu.cn。 马元魁，女，山东青岛人，教授，从事数论及其应用研究，mayuankui@xatu.edu.cn。 mayuankui@xatu.edu.cn 1 1 2 3 西安工业大学　基础学院，陕西　西安　710021 西安工业大学　基础学院，陕西　西安　710021 School of Science, Xi'an Technological University, Xi'an 710021, China School of Science, Xi'an Technological University, Xi'an 710021, China 光云大学　数学系，韩国　首尔　139-701 光云大学　数学系，韩国　首尔　139-701 Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea 上海交通大学　数学科学学院，上海　200240 上海交通大学　数学科学学院，上海　200240 School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China 张欢 研究了Dedekind DC和的一个新的推广——poly-Dedekind DC和。Dedekind DC和中的Euler函数替换成了由polylogarithm函数定义的poly-Euler函数。利用polylogarithm函数的定义、第二类Stirling数的定义、Euler多项式的定义，得到了此类poly-Euler函数满足的一些恒等式，包括此类poly-Euler数与第二类Stirling数之间的关系，以及此类poly-Euler多项式与Euler多项式、第二类Stirling数之间的关系，并证明了此poly-Dedekind DC和满足互反关系。正如经典的Dedekind和一样，可进一步探讨其与模形式、 ζ 函数以及三角和的关系。 As a new generalization of the Dedekind type DC sums, the poly-Dedekind type DC sums are introduced, which are obtained from the Dedekind type DC sums by replacing Euler functions by poly-Euler functions of arbitrary indices arising from polylogarithm functions. By using the definitions of polylogarithm functions, Stirling numbers of second kind, and Euler polynomials, several properties of poly-Euler functions are obtained, including the relations between this type of poly-Euler numbers and Stirling numbers of second kind, the relations between this type of poly-Euler polynomials and Euler polynomials and Stirling numbers of second kind. The poly-Dedekind type DC sums are shown to satisfy a reciprocity relation. It can be further explored in connection with modular forms, ζ functions, and trigonometric sums, just as in the cases of Apostol-Dedekind sums. poly-Dedekind DC和 polylogarithm函数 poly-Euler多项式 poly-Dedekind type DC sums polylogarithm functions poly-Euler polynomials 关于Dedekind和以及各种类似和式的研究在解析数论、模形式、特殊函数理论、代数数论、组合几何、拓扑学以及算法复杂性研究中占有举足轻重的位置，并和Dedekind函数的转化公式、类数、格点问题、Dirichlet L -函数的特殊值问题、群在流形上的作用以及随机数生成器等问题密切相关。 Apostol研究了推广的Dedekind和，把Dedekind和中第一类Bernoulli函数替换成任意Bernoulli函数，并得到了一个互反公式 [ 1 1 - 2 2 1-2 ] 。 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884923&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884933&type= 式中： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884929&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884932&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884936&type= p ( x )= B p ( x －[ x ])为Bernoulli函数； B p ( x )为Bernoulli多项式。定义为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884949&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884951&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884952&type= 式中，[ x ]表示不超过 x 的最大整数。 作为Apostol广义Dedekind和的推广，Kim等研究了用第二类poly-Bernoulli函数定义的poly-Dedekind和 [ 3 3 ] 。Ma Y K等研究了用poly-Bernoulli函数定义的poly-Dedekind和 [ 4 4 ] 。 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884955&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884944&type= 式中， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884945&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884962&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884947&type= ，且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884969&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884980&type= 。 Kim研究了Dedekind DC和 [ 5 5 ] http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884971&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884985&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884973&type= 式中， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884975&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884977&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884976&type= p ( x )= E p ( x －[ x ])为Euler函数 [ 1 1 , 5 5 - 7 7 5-7 ] 。 Ma Y K等研究了poly-Dedekind DC和，把Dedekind DC和中Euler函数替换成用polyexpo-nential函数定义的poly-Euler函数 [ 8 8 ] 。 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884998&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885001&type= 式中： h ， m ， p ∈ N ； http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885018&type= 为poly-Euler函数； http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885038&type= 。 Simsek研究了Dedekind DC和的三角函数表示，并得到了它与其它一些函数或者和式的关系 [ 6 6 ] ，如Clausen函数、polylogarithm函数、Hurwitz ζ 函数、Hardy-Berndt和等。此外，Bayad和Simsek研究了推广的Apostol-Dedekind-Rademacher和，推广了经典的Dedekind-Rademacher和，可以用Jacobi模形式、余切函数或Barnes多重zeta函数的特殊值表示 [ 9 9 ] 。他们证明了这些和式满足互反关系，并能从中推导出一些著名的互反关系。本文的目的是探索与模形式、 ζ 函数以及三角和等有关的新的形式的和。 本文进一步推广了Dedekind DC和，研究了poly-Dedekind DC和，用polylogarithm函数定义poly-Euler函数，并证明它满足互反关系。 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885032&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885052&type= 式中： m ， h ， p ∈ N ； m ≡1(mod 2)； h ≡1(mod 2)； k ∈ Z 。 当 k =1时，此互反公式退化成Dedekind DC和的互反公式(见推论2)。 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885055&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885072&type= 式中： m ， h ， p ∈ N ； m ≡1(mod 2)； h ≡1(mod 2)。 预备知识 1 首先回顾一些符号和定义。已知Euler多项式定义如下 [ 1 1 - 19 19 1-19 ] http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885073&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885076&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885078&type= 式中， E n = E n (0)称为Euler数。 由式(6)可以得到 [ 1 1 - 19 19 1-19 ] http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885083&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885099&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885102&type= 易知，Euler数 E 0 =1， E 1 =－ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885104&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885107&type= ， E 2 =0， E 3 = http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885094&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885120&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885117&type= ， E 4 =0， E 5 =－ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885131&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885126&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885124&type= ，…， E 2 k =0， k =1，2，…。 由式(6)还可以得到 E n (1)+ E n =2 δ 0， n ，( n ≥0)， δ n ， k 表示Kronecker符号。 由式(6)可以推出 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885140&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885147&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885149&type= 已知polylogarithm函数定义为 [ 10 10 - 11 11 10-11 ] http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885168&type= 当 k =1时，Li 1 ( x )=－log(1－ x )。 设 n 是非负整数，第二类Stirling数定义为 [ 10 10 , 12 12 ] http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885177&type= 由poly-Euler函数定义的poly-Dedekind DC和 1 对每一个整数 k ，定义poly-Euler多项式如下 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885185&type= 当 x =0， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885202&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885223&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885219&type= ，称为poly-Euler数。易见，对于任意的 n ≥0, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885216&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885230&type= 。由式（11）容易得到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885252&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885250&type= ， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885239&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885258&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885256&type= ，…，以及 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885260&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885266&type= 把Dedekind DC和中Euler函数替换成用polylogarithm函数定义的poly-Euler函数，定义Poly-Dedekindd DC和为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885268&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885282&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885285&type= 式中： h ， m ， p ∈ N ； http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885273&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885291&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885275&type= 为poly-Euler函数； http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885296&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885300&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885312&type= 。 由式（11）有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885304&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885323&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885322&type= 因此，可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885341&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885348&type= 另一方面， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885354&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885357&type= 因此，由式(15)和式(16)可得如下定理。 定理1 　对于任意 n ∈ N ，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885364&type= 令 k =1，由定理1可得如下推论。 推论1 　对于任意 n ∈ N ，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885378&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885385&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885387&type= 式中 δ n,k 表示Kronecker符号。 接下来的引理同文献[ 8 8 ]中的引理1、引理2、定理7及定理8。 引理1 　对于 p , s ∈ N 且 s < p， 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885401&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885390&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885405&type= 引理2 　对于 p ∈ N ，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885410&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885412&type= 引理3 　对于 m ∈ N 且 m ≡1 (mod 2)， p ≡1 (mod 2)且 p > 1，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885420&type= 引理4 　对于 h ， m ， p ∈ N 且（ h ， m ）=1， m ≡1 (mod 2)，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885422&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885425&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885440&type= 对于 d ∈ N 且 d ≡1 (mod 2)，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885447&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885459&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885451&type= 因此，由式(17)可以得到如下定理。 定理2 　对于 k ∈ Z ， d ∈ N 且 d ≡1 (mod 2), n ≥0，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885453&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885469&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885456&type= 对于 m ， h ∈ N 且 m ≡1(mod 2)， h ≡1(mod 2)，由式(13)和定理2，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885457&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885490&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885493&type= 因此，由式(18)可以得到poly-Dedekind DC和的互反公式如下。 定理3 　对于 m ， h ， p ∈ N ， k ∈ Z 且 m ≡1 (mod 2)， h ≡1 (mod 2)，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885482&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885485&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885487&type= 当 k =1，由推论1可知，以上定理可以退化成Dedekind DC和的互反公式。 推论2 　对于 m ， h ， p ∈ N 且 m ≡1 (mod 2)， h ≡1 (mod 2)，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885506&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885518&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885510&type= 结语 1 Dedekind和及其推广是由Bernoulli函数及其推广定义的，Dedekind DC和及其推广是由Euler函数及其推广定义的，而且它们都满足互反关系。作为Dedekind DC和的进一步推广，本文研究了poly-Dedekind DC和，由polylogarithm函数定义了poly-Euler函数，并证明了它满足互反关系。 参考文献 APOSTOL T M . Generalized Dedekind sums and transformation formulae of certain Lambert series [J]. Duke Math J , 1950 , 17 ( 2 ): 147 - 157 . Generalized Dedekind sums and transformation formulae of certain Lambert series APOSTOL T M . A study of Dedekind sums and their generalization [D]. Berkeley : University of California , 1949 . A study of Dedekind sums and their generalization KIM T , KIM D S , LEE H , et al . Identities on poly-Dedekind sums [J]. Adv Difference Equ , 2020 , 2020 : 563 . Identities on poly-Dedekind sums MA Y K , KIM D S , LEE H , et al . Poly-Dedekind sums associated with poly-Bernoulli functions [J]. J Inequal Appl , 2020 , 2020 : 248 . Poly-Dedekind sums associated with poly-Bernoulli functions KIM T . Note on Dedekind type DC sums [J]. Adv Stud Contemp Math , 2009 , 18 ( 2 ): 249 - 260 . Note on Dedekind type DC sums SIMSEK Y . Special functions related to Dedekind-type DC-sums and their applications [J]. Russ J Math Phys , 2010 , 17 ( 4 ): 495 - 508 . 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