组合邻差方法及其应用
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摘要
组合邻差方法(The method of combinatorial telescoping)是一种证明和发现q-恒等式的有效方法。本文通过对组合邻差方法的深入研究,得到一些新的结果：将组合邻差方法从交错和的情形推广到非交错和的情形：进一步的,将组合邻差方法推广到多币和上使其能够适用于多重和的组合恒等式。
本文共分为六章。第一章主要介绍分拆理论的背景、基本概念以及常川的定义和符号。同时,我们介绍分拆理论的研究中两个常川的工具：分拆的图像表示和分拆的生成函数。本章最后简要概括了全文的内容和结构。
在第二章中,我们回顾了组合邻差方法及其背景。在Zeilberger算法的基础上,陈水川、侯庆虎和孙慧提出了组合邻差方法,该方法被广泛应用在许多组合恒等式的证明中,包括Watson等式,Sylvester等式等。我们将用组合邻差方法从递推关系的角度证明Ramanujan's Lost Notebook中的一个部分theta等式。
在第三章中,我们将组合邻差方法推广到非交错和上,并利用这一推广解决了Andrews提出的一个公开问题。原始的组合邻差方法只考虑了交错和的情况,我们考虑了非交错和的情况,即求和项中不含有(-1)k作为因子的组合恒等式。作为应用,我们从新的角度证明了一系列经典的q-恒等式,包括q-二项式定理,MacMahon等式,Lebesgue等式以及Yee等式。特别的,Andrews用代数方法证明了一个与q-雅可比多项式有关的等式,并将其组合证明列为一个公开问题,我们将用组合邻差方法从递推关系的角度给出一个组合证明。
在第四章中,基于多重和的Zeilberger算法,我们可将组合邻差方法推广到多重和情形,本章中给出了双重和情形下组合邻筹方法的详细描述。作为应用,我们将考虑Andrews提出的两个恒等式,并利用组合邻差方法给出这两个等式的推广形式。
在第章章中,我们将由组合邻差方法引导出q-恒等式的直接组合证明。具体来说,通过组合邻差方法,我们可以得到双射φ:A∪H→B∪H,进一步考虑Feldman和Propp提出的消去法,对Φ进行迭代直到A中所有元素的像落在B中,即可得到双射Ψ：A→B。这一思路可以用来证明许多组合恒等式。作为例子,我们将给出一个q-恒等式的直接组合证明。
在第六章中,我们考虑不同的奇偶性参数在分拆恒等式中的体现。奇偶性是分拆理论研究中的一个重要的指标。欧拉定理作为分拆理论的基石,指出把n分成不同部分的分拆的个数等于n的奇分拆的个数。Andrews在将不同的奇偶性参数引入分拆的过程中得到了一系列分拆恒等式,我们借助德菲方块(Durfee Square)给出其中一个的组合证明。
The method of combinatorial telescoping is useful for proving and discovering q-series identities. The main object of this thesis is to generalize this method. We provide the method of combinatorial telescoping for sums of positive terms, while the original method can only be applied to alternating sums. Moreover, in light of the creative telescoping method for multiple sums, we also present the method of combinatorial telescoping for multiple sums.
This thesis is organized as follows. The first chapter is devoted to an introduction to the background, the basic concepts and the usual notations of partition theory. We will present two of the most elemental tools for treating partitions:graphical represen-tation of partitions; infinite product generating functions. The outline of this thesis will be presented at the end of this chapter.
Chapter2is devoted to the method of combinatorial telescoping presented by Chen, Hou and Sun. This method is useful for proving q-series identities, such as Watson's identity and Sylvester's identity. With the aid of combinatorial telescoping we shall describe a new proof of a partial theta series identity from Ramanujan's Lost Notebook.
In Chapter3, we will present the method of combinatorial telescoping for sums of positive terms, which is a variant of the method of combinatorial telescoping for alternating sums. In this chapter, we shall describe the method in detail and apply it to identities whose summands do not have the factor (-1)k. We illustrate this idea by giving a new way to prove the classical identities, such as the q-binomial theorem, MacMahon's identity, Lebesgue's identity as well as two identities of Yee. Besides, Andrews proposed a problem of finding a combinatorial proof of an identity on the q-little Jacobi polynomials. By the method of combinatorial telescoping for sums of positive terms, we establish a recurrence relation that leads to the identity of Andrews.
In Chapter4, we will generalize the method of combinatorial telescoping such that it can be applied to multiple sums. We shall show the construction of the combinatorial telescoping for double sums. To illustrate this method, we shall consider two identities of Andrews. With the aid of combinatorial telescoping for double sums, we are able to prove two beautiful identities that reduce to the identities of Andrews.
In Chapter5, we present a way to find a direct combinatorial interpretation via the method of combinatorial telescoping. The method of combinatorial telescoping gives an explanation of a recurrence relation, namely,0:A∪H→B∪H. By the method of cancelation presented by Feldman and Propp, the bijection φ implies a bijection ψ:A→B. More precisely, we can define the bijection ψ:A→B by setting ψ (a) to be the first element b that falls into B while iterating the action of φ on a∈A. We shall illustrate this idea by describing an involution, which leads to a proof of an identity of Andrews.
Chapter6is mainly focus on parity in partition identities. Parity has played an important role in partition identities. Euler's partition theorem states that the number of partitions of n into odd parts equals to the number of partitions of n into distinct parts. Andrews investigated a variety of parity questions in partition identities. We state an identity of Andrews, for which he ask for a combinatorial proof and we shall prove it combinatorially with the aid of Durfee square.
本文共分为六章。第一章主要介绍分拆理论的背景、基本概念以及常川的定义和符号。同时,我们介绍分拆理论的研究中两个常川的工具：分拆的图像表示和分拆的生成函数。本章最后简要概括了全文的内容和结构。
在第二章中,我们回顾了组合邻差方法及其背景。在Zeilberger算法的基础上,陈水川、侯庆虎和孙慧提出了组合邻差方法,该方法被广泛应用在许多组合恒等式的证明中,包括Watson等式,Sylvester等式等。我们将用组合邻差方法从递推关系的角度证明Ramanujan's Lost Notebook中的一个部分theta等式。
在第三章中,我们将组合邻差方法推广到非交错和上,并利用这一推广解决了Andrews提出的一个公开问题。原始的组合邻差方法只考虑了交错和的情况,我们考虑了非交错和的情况,即求和项中不含有(-1)k作为因子的组合恒等式。作为应用,我们从新的角度证明了一系列经典的q-恒等式,包括q-二项式定理,MacMahon等式,Lebesgue等式以及Yee等式。特别的,Andrews用代数方法证明了一个与q-雅可比多项式有关的等式,并将其组合证明列为一个公开问题,我们将用组合邻差方法从递推关系的角度给出一个组合证明。
在第四章中,基于多重和的Zeilberger算法,我们可将组合邻差方法推广到多重和情形,本章中给出了双重和情形下组合邻筹方法的详细描述。作为应用,我们将考虑Andrews提出的两个恒等式,并利用组合邻差方法给出这两个等式的推广形式。
在第章章中,我们将由组合邻差方法引导出q-恒等式的直接组合证明。具体来说,通过组合邻差方法,我们可以得到双射φ:A∪H→B∪H,进一步考虑Feldman和Propp提出的消去法,对Φ进行迭代直到A中所有元素的像落在B中,即可得到双射Ψ：A→B。这一思路可以用来证明许多组合恒等式。作为例子,我们将给出一个q-恒等式的直接组合证明。
在第六章中,我们考虑不同的奇偶性参数在分拆恒等式中的体现。奇偶性是分拆理论研究中的一个重要的指标。欧拉定理作为分拆理论的基石,指出把n分成不同部分的分拆的个数等于n的奇分拆的个数。Andrews在将不同的奇偶性参数引入分拆的过程中得到了一系列分拆恒等式,我们借助德菲方块(Durfee Square)给出其中一个的组合证明。
The method of combinatorial telescoping is useful for proving and discovering q-series identities. The main object of this thesis is to generalize this method. We provide the method of combinatorial telescoping for sums of positive terms, while the original method can only be applied to alternating sums. Moreover, in light of the creative telescoping method for multiple sums, we also present the method of combinatorial telescoping for multiple sums.
This thesis is organized as follows. The first chapter is devoted to an introduction to the background, the basic concepts and the usual notations of partition theory. We will present two of the most elemental tools for treating partitions:graphical represen-tation of partitions; infinite product generating functions. The outline of this thesis will be presented at the end of this chapter.
Chapter2is devoted to the method of combinatorial telescoping presented by Chen, Hou and Sun. This method is useful for proving q-series identities, such as Watson's identity and Sylvester's identity. With the aid of combinatorial telescoping we shall describe a new proof of a partial theta series identity from Ramanujan's Lost Notebook.
In Chapter3, we will present the method of combinatorial telescoping for sums of positive terms, which is a variant of the method of combinatorial telescoping for alternating sums. In this chapter, we shall describe the method in detail and apply it to identities whose summands do not have the factor (-1)k. We illustrate this idea by giving a new way to prove the classical identities, such as the q-binomial theorem, MacMahon's identity, Lebesgue's identity as well as two identities of Yee. Besides, Andrews proposed a problem of finding a combinatorial proof of an identity on the q-little Jacobi polynomials. By the method of combinatorial telescoping for sums of positive terms, we establish a recurrence relation that leads to the identity of Andrews.
In Chapter4, we will generalize the method of combinatorial telescoping such that it can be applied to multiple sums. We shall show the construction of the combinatorial telescoping for double sums. To illustrate this method, we shall consider two identities of Andrews. With the aid of combinatorial telescoping for double sums, we are able to prove two beautiful identities that reduce to the identities of Andrews.
In Chapter5, we present a way to find a direct combinatorial interpretation via the method of combinatorial telescoping. The method of combinatorial telescoping gives an explanation of a recurrence relation, namely,0:A∪H→B∪H. By the method of cancelation presented by Feldman and Propp, the bijection φ implies a bijection ψ:A→B. More precisely, we can define the bijection ψ:A→B by setting ψ (a) to be the first element b that falls into B while iterating the action of φ on a∈A. We shall illustrate this idea by describing an involution, which leads to a proof of an identity of Andrews.
Chapter6is mainly focus on parity in partition identities. Parity has played an important role in partition identities. Euler's partition theorem states that the number of partitions of n into odd parts equals to the number of partitions of n into distinct parts. Andrews investigated a variety of parity questions in partition identities. We state an identity of Andrews, for which he ask for a combinatorial proof and we shall prove it combinatorially with the aid of Durfee square.
引文
[1]K. Alladi and B. Gordon, Partition identities and a continued fraction of Ramanu-jan, J. Combin. Theory Ser. A,63 (1993),275-300.
[2]K. Alladi, A partial theta identity of Ramanujan and its number-theoretic inter-pretation, Ramanujan J.,20 (2009),329-339
[3]G.E. Andrews, An analytic proof of the Rogers-Ramanujan-Gordon identities, Amer. J. Math.,88 (1966),844-846.
[4]G.E. Andrews, An analytic generalization of the Rogers-Ramanujan identities for odd moduli, Proc. Natl. Acad. Sci.,71 (1974),4082-4085.
[5]G.E. Andrews, Partially ordered sets and the Rogers-Ramanujan identities, Ae-quationes Math.,12 (1975),94-107.
[6]G.E. Andrews,q-Series:Their Development and Application in Analysis, Num-ber Theory, Combinatorics, Physics, and Computer Algebra, Conf. Board Math. Sci.,66,1986.
[7]G.E. Andrews, The Theory of Partitions, Cambridge University Press, Cambridge, 1998.
[8]G.E. Andrews, Parity in partition identities, Ramanujan J.,23 (2010),45-90.
[9]G.E. Andrews and B.C. Berndt, Ramanujan's Lost Notebook:Part II, Springer, New York,2009.
[10]R. Askey, Ramanujan's extensions of the gamma and beta function, Amer. Math. Monthly,87 (1980),346-359.
[11]B.C. Berndt, B. Kim and A.J. Yee, Ramanujan's lost notebook:combinatorial proofs of identities associated with Heine's transformation or partial.theta func-tions, J. Combin. Theory Ser. A,117 (2010),957-973.
[12]C. Bessenrodt, A bijection for Lebesgue's partition identity in the spirit of Sylvester, Discrete Math.,132 (1994),1-10.
[13]C. Boulet, Partition identity bijections related to sign-balance and rank, Ph.D. thesis, MIT,2005.
[14]D.M. Bressoud, A simple proof of Mehler's formula for q-Hermite polynomials, Indiana Univ. Math. J,29 (1980),577-580.
[15]D. Bressoun, Proofs and confirmations, the story of the alternating sign matrix conjecture, Cambridge University Press, Cambridge,1999.
[16]A.L. Cauchy, Memoire sur les Fonctions dont plusiers valeurs, CR Acad Sci. Paris,17 (1843),526-534.
[17]W.Y.C. Chen, D.K. Du and C.B. Mei, Combinatorial telescoping for an identity of Andrews on parity in partitions, European J. Combin.,33 (2012),510-518.
[18]W.Y.C. Chen, Q.H. Hou and Y.P. Mu, A telescoping method for double summa-tions, J. Comput. Appl. Math.,196 (2006),553-566.
[19]W.Y.C. Chen, Q.H. Hou and L.H. Sun, The method of combinatorial telescoping, J. Combin. Theory Ser. A,118 (2011),899-907.
[20]W.Y.C. Chen, Q.H. Hou, and L.H. Sun, An iterated map for the Lebesgue identity, preprint.
[21]W.Y.C. Chen and E.H. Liu, A Franklin type involution for squares, Adv. in Appl. Math., (2012).
[22]F.J. Dyson, A new symmetry of partitions, J. Combin. Theory Ser. A,7 (1969), 56-61.
[23]L. Euler, Introductio analysin infinitorum, Lausanne,1 (1748),253-275.
[24]M.C. Fasenmyer, Some generalized hypergeometric polynomials, Ph.D. thesis, University of Michigan,1945.
[25]M.C. Fasenmyer, Some generalized hypergeometric polynomials, Bull. Amer. Math. Soc.,53 (1947),806-812.
[26]D. Feldman and J. Propp, Producing new bijections from old, Adv. Math.,113 (1995),1-44.
[27]N.J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988.
[28]F. Franklin, Sur le developpement du produit infini (1-x)(1-x2)(1-x3)(1-x4).., CR Acad. Sci. Paris,92 (1881),448-450.
[29]A.M. Fu, A combinatorial proof of the Lebesgue identity, Discrete Math.,308 (2008),2611-2613.
[30]R. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge,1990.
[31]H. Gollnitz, Einfache partionen, Diplomarbeit WS, Gottingen,1960.
[32]H. Gollnitz, Partitionen mit differenzenbedingungen, J. Reine Angew. Math.,225 (1967),154-190.
[33]J.W.L Glaisher, A theorem in partitions, Messenger of Math.,12(1883),158-170.
[34]B. Gordon, A combinatorial generalization of the Rogers-Ramanujan identities, Amer. J. Math.,83 (1961),393-399.
[35]B. Gordon, Some Ramanujan-like continued fractions, Short Communications, Int. Congr. of Math., (1962),29-30.
[36]B. Gordon, Some continued fractions of the Rogers-Ramanujan type, Duke Math. J.,31 (1965),741-748.
[37]R. Graham, D.E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA,1994.
[38]E. Heine, Untersuchungen uber die Reihe 1+...x2+..., J. Reine Angew. Math., 34(1847),285-328.
[39]D. Kim and A.J. Yee, A note on partitions into distinct parts and odd parts, Ra-manujan J.,3 (1999),227-231.
[40]T.H. Koornwinder, On Zeilberger's algorithm and its q-analogue, J. Comput. Ap-pl. Math.,48 (1993),91-111.
[41]A. Lascoux, Sylvester's bijection between strict and odd partitions, Discrete Math.,277 (2004),275-278.
[42]D.P. Little and J.A. Sellers, New proofs of identities of Lebesgue and Gollnitz via tilings, J. Combin. Theory Ser. A,116 (2009),223-231.
[43]P.A. MacMahon, Combinatory Analysis, Cambridge University Press, Cam-bridege,1916.
[44]M. Mohammed and D. Zeilberger, Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening of Wilf-Zeilberger theory, Adv. in Appl. Math.,37 (2006),139-152.
[45]I. Pak, On Fine's partition theorem, Dyson, Andrews, and missed opportunities. Math. Intelligencer,25 (2003),10-34.
[46]I. Pak, Partition bijections, a survey, Ramanujan J.,12 (2006),5-75.
[47]I. Pak and A. Postnikov, A generalization of Sylvester's identity, Discrete Math., 178 (1998),277-281.
[48]P. Paule and A. Riese, A mathematica q-analogue of Zeilberger's algorithm based on an algebraically motivated approach to q-hypergeometric telescoping, In: "Special Functions, q-Series and Related Topics", Fields Inst. Commun.,14 (1997),179-210.
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[51]S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi,1988.
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[62]A.J. Yee, Ramanujan's partial theta series and parity in partitions, Ramanujan J., 23(2010),215-225.
[63]D. Zeilberger, Sister Celine's technique and its generalizations, J. Math. Anal. Appl.,85(1982),114-145.
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[65]D. Zeilberger, A fast algorithm for proving terminating hypergeometric identities, Discrete Math.,80 (1990),207-211.
[66]D. Zeilberger, The method of creative telescoping, J. Symbolic Comput.,11 (1991),195-204.
[2]K. Alladi, A partial theta identity of Ramanujan and its number-theoretic inter-pretation, Ramanujan J.,20 (2009),329-339
[3]G.E. Andrews, An analytic proof of the Rogers-Ramanujan-Gordon identities, Amer. J. Math.,88 (1966),844-846.
[4]G.E. Andrews, An analytic generalization of the Rogers-Ramanujan identities for odd moduli, Proc. Natl. Acad. Sci.,71 (1974),4082-4085.
[5]G.E. Andrews, Partially ordered sets and the Rogers-Ramanujan identities, Ae-quationes Math.,12 (1975),94-107.
[6]G.E. Andrews,q-Series:Their Development and Application in Analysis, Num-ber Theory, Combinatorics, Physics, and Computer Algebra, Conf. Board Math. Sci.,66,1986.
[7]G.E. Andrews, The Theory of Partitions, Cambridge University Press, Cambridge, 1998.
[8]G.E. Andrews, Parity in partition identities, Ramanujan J.,23 (2010),45-90.
[9]G.E. Andrews and B.C. Berndt, Ramanujan's Lost Notebook:Part II, Springer, New York,2009.
[10]R. Askey, Ramanujan's extensions of the gamma and beta function, Amer. Math. Monthly,87 (1980),346-359.
[11]B.C. Berndt, B. Kim and A.J. Yee, Ramanujan's lost notebook:combinatorial proofs of identities associated with Heine's transformation or partial.theta func-tions, J. Combin. Theory Ser. A,117 (2010),957-973.
[12]C. Bessenrodt, A bijection for Lebesgue's partition identity in the spirit of Sylvester, Discrete Math.,132 (1994),1-10.
[13]C. Boulet, Partition identity bijections related to sign-balance and rank, Ph.D. thesis, MIT,2005.
[14]D.M. Bressoud, A simple proof of Mehler's formula for q-Hermite polynomials, Indiana Univ. Math. J,29 (1980),577-580.
[15]D. Bressoun, Proofs and confirmations, the story of the alternating sign matrix conjecture, Cambridge University Press, Cambridge,1999.
[16]A.L. Cauchy, Memoire sur les Fonctions dont plusiers valeurs, CR Acad Sci. Paris,17 (1843),526-534.
[17]W.Y.C. Chen, D.K. Du and C.B. Mei, Combinatorial telescoping for an identity of Andrews on parity in partitions, European J. Combin.,33 (2012),510-518.
[18]W.Y.C. Chen, Q.H. Hou and Y.P. Mu, A telescoping method for double summa-tions, J. Comput. Appl. Math.,196 (2006),553-566.
[19]W.Y.C. Chen, Q.H. Hou and L.H. Sun, The method of combinatorial telescoping, J. Combin. Theory Ser. A,118 (2011),899-907.
[20]W.Y.C. Chen, Q.H. Hou, and L.H. Sun, An iterated map for the Lebesgue identity, preprint.
[21]W.Y.C. Chen and E.H. Liu, A Franklin type involution for squares, Adv. in Appl. Math., (2012).
[22]F.J. Dyson, A new symmetry of partitions, J. Combin. Theory Ser. A,7 (1969), 56-61.
[23]L. Euler, Introductio analysin infinitorum, Lausanne,1 (1748),253-275.
[24]M.C. Fasenmyer, Some generalized hypergeometric polynomials, Ph.D. thesis, University of Michigan,1945.
[25]M.C. Fasenmyer, Some generalized hypergeometric polynomials, Bull. Amer. Math. Soc.,53 (1947),806-812.
[26]D. Feldman and J. Propp, Producing new bijections from old, Adv. Math.,113 (1995),1-44.
[27]N.J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988.
[28]F. Franklin, Sur le developpement du produit infini (1-x)(1-x2)(1-x3)(1-x4).., CR Acad. Sci. Paris,92 (1881),448-450.
[29]A.M. Fu, A combinatorial proof of the Lebesgue identity, Discrete Math.,308 (2008),2611-2613.
[30]R. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge,1990.
[31]H. Gollnitz, Einfache partionen, Diplomarbeit WS, Gottingen,1960.
[32]H. Gollnitz, Partitionen mit differenzenbedingungen, J. Reine Angew. Math.,225 (1967),154-190.
[33]J.W.L Glaisher, A theorem in partitions, Messenger of Math.,12(1883),158-170.
[34]B. Gordon, A combinatorial generalization of the Rogers-Ramanujan identities, Amer. J. Math.,83 (1961),393-399.
[35]B. Gordon, Some Ramanujan-like continued fractions, Short Communications, Int. Congr. of Math., (1962),29-30.
[36]B. Gordon, Some continued fractions of the Rogers-Ramanujan type, Duke Math. J.,31 (1965),741-748.
[37]R. Graham, D.E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA,1994.
[38]E. Heine, Untersuchungen uber die Reihe 1+...x2+..., J. Reine Angew. Math., 34(1847),285-328.
[39]D. Kim and A.J. Yee, A note on partitions into distinct parts and odd parts, Ra-manujan J.,3 (1999),227-231.
[40]T.H. Koornwinder, On Zeilberger's algorithm and its q-analogue, J. Comput. Ap-pl. Math.,48 (1993),91-111.
[41]A. Lascoux, Sylvester's bijection between strict and odd partitions, Discrete Math.,277 (2004),275-278.
[42]D.P. Little and J.A. Sellers, New proofs of identities of Lebesgue and Gollnitz via tilings, J. Combin. Theory Ser. A,116 (2009),223-231.
[43]P.A. MacMahon, Combinatory Analysis, Cambridge University Press, Cam-bridege,1916.
[44]M. Mohammed and D. Zeilberger, Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening of Wilf-Zeilberger theory, Adv. in Appl. Math.,37 (2006),139-152.
[45]I. Pak, On Fine's partition theorem, Dyson, Andrews, and missed opportunities. Math. Intelligencer,25 (2003),10-34.
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