三类条件分拆函数同余性质的研究
发布时间：2018-07-26 20:40
【摘要】：条件分拆函数的同余性质是当前组合数学研究的热点问题之一,它与q-级数、数论、代数学、机器证明等多个数学分支有着广泛而密切的联系,并在数学、物理、概率论、计算机科学等领域有着重要的应用。近年来,数学研究人员虽然发现了许多条件分拆函数的同余关系,但仍有许多问题有待进一步研究解决。本文主要研究了广义Frobenius-6着色分拆函数,条件Binary分拆函数和Overpartitions分拆函数的同余性质,具体工作如下:在第一章中,介绍了条件分拆函数的研究背景,研究进展以及本文的研究内容。在第二章中,我们借助Hirschhorn给出的)13(6fnc(10)的生成函数,并利用分块公式和q-级数运算,证明了Baruah和Sarmah教授提出的一个关于广义Frobenius-6着色分拆函数)(6fnc模243的同余关系的猜想。在此基础上,我们还建立了若干新的关于广义Frobenius-6着色分拆函数)(6fnc模3的更高次幂的同余关系。在第三章中,利用代数组合方法和q-级数运算,我们证明了大量的关于Ramanujan型条件Binary分拆函数nW)(模2和3的高次幂的同余关系,从而解决了Lan和Sellers教授提出的一个公开问题。在第四章中,我们先利用计算机代数方法和theta函数恒等式建立了(?)(5n)的生成函数,然后利用二次剩余理论建立了若干新的关于Overpartitions分拆函数(?)(n)(模5和9的无穷族同余关系,推广了Treneer以及Chen,Sun,Wang和Zhang给出的结论。
[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.
【学位授予单位】：江苏大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157
本文编号：2147233
[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.
【学位授予单位】：江苏大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157
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