發(fā)布時間：2018-04-21 13:11

  本文選題：混合數(shù) + theta函數(shù)　； 參考：《江蘇大學(xué)》2017年碩士論文

【摘要】：正整數(shù)表示為多個混合數(shù)之和的表示方法數(shù)是當(dāng)前組合數(shù)學(xué)和數(shù)論領(lǐng)域的研究熱點(diǎn)之一,該課題與多個數(shù)學(xué)分支有著重要的聯(lián)系,吸引了包括高斯在內(nèi)的眾多數(shù)學(xué)研究者的興趣。本文主要研究了正整數(shù)表示為六個變量和八個變量的混合數(shù)之和的表示方法數(shù)。令N和(r,s;t;n)和N(a1,a2,a3;b1,b2;c1,c2,c3.c4;n)分別為將正整數(shù)n表示成含六個未知數(shù)和含八個未知數(shù)之和的表示方法數(shù)。在本文中,對于任意的正整數(shù)n和某些特殊的{r,s,t}、{a1,a2,a3;b1,b2;c1,c2,c3,c4}、利用theta函數(shù)和Eisenstein級數(shù)的(p,k)參數(shù)公式,確定了一些N(r,s;t;n),N(a1,a2,a3;b1,b2;c1,c2,c3,c4;n)的精確公式。本文結(jié)構(gòu)如下:在第一章中,主要介紹了研究背景、研究現(xiàn)狀以及研究的主要內(nèi)容。第二章中,主要介紹了 theta函數(shù)、Eisenstein級數(shù)及它們的(p,k)參數(shù)表示,為第三章和第四章中建立新的包含theta函數(shù)和Eisenstein級數(shù)的恒等式做好準(zhǔn)備。在第三章中,利用theta函數(shù)的(p,k)參數(shù)表示建立了一系列新的theta函數(shù)恒等式,研究了將正整數(shù)n表示為六個變量的混合數(shù)之和的表示方法數(shù),確定了N(2,0;1;n),N(2,0;2;n),N(2,0;4;n),N(1,1;1;n),N(1,1;2;n),N(1,1;4;n),N(0,2;1;n),N(0,2;2;n)和N(0,2;4;n)的精確公式。在第四章中,引入了函數(shù)G(q和H(q),利用Eisenstein級數(shù)的(p,k)參數(shù)表示,建立了多個包含Eisenstein級數(shù)的恒等式,研究了將正整數(shù)n表示為八個變量的混合數(shù)之和的表示方法數(shù),刻畫出了某些N(a1,a2,a;b1,b2;c1,c2,c3,c4;n)的精確公式。最后,我們對研究工作進(jìn)行了總結(jié)并展望后續(xù)的研究工作。
[Abstract]:The representation of positive integers as the sum of multiple mixed numbers is one of the hot topics in the field of combinatorial mathematics and number theory, which has an important relationship with many branches of mathematics. Attracted the interest of many mathematics researchers, including Gao Si. In this paper, we study the representation of positive integers as the sum of mixed numbers of six variables and eight variables. The results show that N and RX ~ (2) and N ~ (1) A ~ (2) A ~ (2) A ~ (3) B ~ (1) B ~ (2 +) C ~ (2 +) C ~ (2) C ~ (2) C ~ (3) C ~ (2) C ~ (3) C ~ (4) ~ (n)) are the method numbers for representing a positive integer n as the sum of six unknowns and eight unknowns, respectively. In this paper, for any positive integer n and some special {rrstt}, {a1a ~ 2a ~ (2) B ~ (1) B ~ (1) B ~ (2 +) C ~ (1) C ~ (2 +) C ~ (3) C ~ (3) C ~ (3) C ~ (4), using the theta function and the formula of Eisenstein series, some exact formulas of Nrs _ t _ t _ n _ n ~ (1) ~ a ~ (2) A ~ (3) B ~ (1) B ~ (1) C ~ (1) C ~ (2) C ~ (3) C ~ (3) C ~ (4) ~ (?)) are determined by using the formula of theta's function and the parameter of Eisenstein series. The structure of this paper is as follows: in the first chapter, the research background, research status and main contents are introduced. In the second chapter, we mainly introduce the theta function Eisenstein series and their parameter representations, and prepare for the establishment of new identities including theta function and Eisenstein series in Chapter 3 and Chapter 4. 鍦ㄧ��涓夌珷涓，

本文編號：1782574