發(fā)布時(shí)間：2018-07-26 20:40

【摘要】：條件分拆函數(shù)的同余性質(zhì)是當(dāng)前組合數(shù)學(xué)研究的熱點(diǎn)問(wèn)題之一,它與q-級(jí)數(shù)、數(shù)論、代數(shù)學(xué)、機(jī)器證明等多個(gè)數(shù)學(xué)分支有著廣泛而密切的聯(lián)系,并在數(shù)學(xué)、物理、概率論、計(jì)算機(jī)科學(xué)等領(lǐng)域有著重要的應(yīng)用。近年來(lái),數(shù)學(xué)研究人員雖然發(fā)現(xiàn)了許多條件分拆函數(shù)的同余關(guān)系,但仍有許多問(wèn)題有待進(jìn)一步研究解決。本文主要研究了廣義Frobenius-6著色分拆函數(shù),條件Binary分拆函數(shù)和Overpartitions分拆函數(shù)的同余性質(zhì),具體工作如下:在第一章中,介紹了條件分拆函數(shù)的研究背景,研究進(jìn)展以及本文的研究?jī)?nèi)容。在第二章中,我們借助Hirschhorn給出的)13(6fnc(10)的生成函數(shù),并利用分塊公式和q-級(jí)數(shù)運(yùn)算,證明了Baruah和Sarmah教授提出的一個(gè)關(guān)于廣義Frobenius-6著色分拆函數(shù))(6fnc模243的同余關(guān)系的猜想。在此基礎(chǔ)上,我們還建立了若干新的關(guān)于廣義Frobenius-6著色分拆函數(shù))(6fnc模3的更高次冪的同余關(guān)系。在第三章中,利用代數(shù)組合方法和q-級(jí)數(shù)運(yùn)算,我們證明了大量的關(guān)于Ramanujan型條件Binary分拆函數(shù)nW)(模2和3的高次冪的同余關(guān)系,從而解決了Lan和Sellers教授提出的一個(gè)公開問(wèn)題。在第四章中,我們先利用計(jì)算機(jī)代數(shù)方法和theta函數(shù)恒等式建立了(?)(5n)的生成函數(shù),然后利用二次剩余理論建立了若干新的關(guān)于Overpartitions分拆函數(shù)(?)(n)(模5和9的無(wú)窮族同余關(guān)系,推廣了Treneer以及Chen,Sun,Wang和Zhang給出的結(jié)論。
[Abstract]:The congruence property of conditional partition function is one of the hot issues in combinatorial mathematics. It has extensive and close relations with many branches of mathematics, such as q-series, number theory, algebra, machine proving, and so on, and is widely and closely related in mathematics, physics, probability theory, etc. Computer science and other fields have important applications. In recent years, although mathematical researchers have discovered many congruence relations of conditional partition functions, there are still many problems to be solved. In this paper, the congruence properties of generalized Frobenius-6 coloring partition function, conditional Binary partition function and Overpartitions partition function are studied. The main work is as follows: in the first chapter, the background of conditional partition function is introduced. Research progress and research content of this paper. In the second chapter, we prove a conjecture about the congruence relation of generalized Frobenius-6 coloring partition function (6fnc module 243) proposed by Professor Baruah and Sarmah by means of the generating function of 6fnc (10) (given by Hirschhorn), and by using block formula and q-series operation. On this basis, we also establish some new congruence relations for the higher power of the generalized Frobenius-6 coloring partition function (6fnc module 3). In chapter 3, by using the algebraic combination method and q-series operations, we prove a large number of congruence relations on the higher power of the nW) (module 2 and 3 of the conditional Binary partition function of Ramanujan type, thus solving an open problem put forward by professors Lan and Sellers. In chapter 4, we first establish the generating function of (?) (5n) by using the computer algebra method and the identity of theta function, then we establish some new infinite family congruence relations about the Overpartitions partition function (?) (n) (module 5 and 9) by using the quadratic residue theory. The conclusions given by Treneer, Chenan Sunn Wang and Zhang are generalized.
【學(xué)位授予單位】：江蘇大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O157

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