Riordan群综述
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摘要
本文主要就Lagrange反演公式、Fa(?) di Bruno公式和Riordan群各自理论形成、内容方法以及彼此之间的联系和区别所做的一个综述。
第一章总结了Riordan群的各种主要定义以及与它相关的Lagrang反演公式和Fa(?) di Bruno公式。
第二章对Riordan群理论的主要运算规则、目前已经建立的Riordan群的主要理论做一概述,其中由Lagrange反演公式给出的运算法则对级数求和有着重要作用。
第三章主要分析了Riordan群在计算组合和式方面以及反演关系方面的应用,并且纠正了Egorychev等人文章中的一个组合反演关系,推广了H.Wilf的著作《Generating Functionology》中一个问题结论。
第四章通过详细计算阐述了这样一个历史误会(这也是本篇综述的目的所在):Riordan群不是人们理解成的一个新方法,其思想早就蕴含在一个多世纪前就存在的Fa(?) di Bruno公式中了。
最后一章提出了一个尚待解决的问题。
In this thesis,we try to make a comprehensive summary on Riordan group as well as its interplay with the celebrated Lagrange inversion formula,the famous Fa(?) di Bruno formula and their various applications in Combinatorial Analysis.
Chapter one we summarize all main concepts of the Riordan group and Riordan array and its connections with the Lagrange inversion formula Fa(?) di Bruno formula.
Chapter two is devoted to all main results in the theory of the Riordan group obtained up to date,some frequently used techniques invoking the Lagrange inversion formula are also rephrased.
In the third chapter,we skctch three aspects of applications of the Riordan group, more precisely,in combinatorial summations and their dual forms in viewpoint of inverse relations.In addition,we give the correct form of an identity poised by Egorychev et al.with an evident error,and a generalization of an identity recorded by Wilf in his book "Generatingfunctionology".
In Chapter four,as a basic motivation of this thesis,we put forward a novel opinion on a historical misunderstanding by a direct calculation:the Riordan group is just a special case of the Fa(?) di Bruno formula,which can date back to more than one and half century ago.
The final chapter we pose a new kind of recursive relations as an open problem.
第一章总结了Riordan群的各种主要定义以及与它相关的Lagrang反演公式和Fa(?) di Bruno公式。
第二章对Riordan群理论的主要运算规则、目前已经建立的Riordan群的主要理论做一概述,其中由Lagrange反演公式给出的运算法则对级数求和有着重要作用。
第三章主要分析了Riordan群在计算组合和式方面以及反演关系方面的应用,并且纠正了Egorychev等人文章中的一个组合反演关系,推广了H.Wilf的著作《Generating Functionology》中一个问题结论。
第四章通过详细计算阐述了这样一个历史误会(这也是本篇综述的目的所在):Riordan群不是人们理解成的一个新方法,其思想早就蕴含在一个多世纪前就存在的Fa(?) di Bruno公式中了。
最后一章提出了一个尚待解决的问题。
In this thesis,we try to make a comprehensive summary on Riordan group as well as its interplay with the celebrated Lagrange inversion formula,the famous Fa(?) di Bruno formula and their various applications in Combinatorial Analysis.
Chapter one we summarize all main concepts of the Riordan group and Riordan array and its connections with the Lagrange inversion formula Fa(?) di Bruno formula.
Chapter two is devoted to all main results in the theory of the Riordan group obtained up to date,some frequently used techniques invoking the Lagrange inversion formula are also rephrased.
In the third chapter,we skctch three aspects of applications of the Riordan group, more precisely,in combinatorial summations and their dual forms in viewpoint of inverse relations.In addition,we give the correct form of an identity poised by Egorychev et al.with an evident error,and a generalization of an identity recorded by Wilf in his book "Generatingfunctionology".
In Chapter four,as a basic motivation of this thesis,we put forward a novel opinion on a historical misunderstanding by a direct calculation:the Riordan group is just a special case of the Fa(?) di Bruno formula,which can date back to more than one and half century ago.
The final chapter we pose a new kind of recursive relations as an open problem.
引文
[1]M.Aigner.Catalan-like numbers and determinants.J.of Combinatorial Theory Ser.A,87 (1999)33-51.
[2]M.Aigner.A characterization of Bell numbers.Discrete Mathematics,205 (1999)207-210.
[3]George E.Andrews,Richard Askey,Ranjan Roy.Special Function,清华大学出版社．2004．
[4]D.Baccherini,D.Mcrlini,and R.Sprugnoli.Level generating trees and proper Riordan arrays.Applicable Analysis and Discrete Mathematics,2 (2008)69-91.
[5]P.Barry.A Catalan transform and related transformations on integer sequences.J.of Integer Sequences,8 (2005).
[6]P.Barry.On a family of generalized Pascal triangles denned by exponential Riordan arrays.J.of Integer Sequences,10 (2007).
[7]P.Barry.Some observations on the Lah and Laguerre transforms of integer sequences.J.of Integer Sequences,10 (2007).
[8]P.Barry.A note on Krawtchouk polynomials and Riordan arrays.J.of Integer Sequences,11 (2008).
[9]P.Barry,and P.Fitzpatrick.On a one-parameter family of Riordan arrays and the weight distribution of MDS codes.J.of Integer Sequences,9 (2007).
[10]N.T.Cameron,and A.Nkwanta.On some (pseudo)involutions in the Riordan group.J.of Integer Sequences,8 (2005).
[11]G.-S.Cheon,and M.E.A.El-Mikkawy.Generalized harmonic numbers with Riordan arrays.J.of Number Theory,128 (2008)413-425.
[12]G.-S.Cheon.M.E.A.El-Mikkawy,and H.-G.Seol.New identities for Stirling numbers via Riordan arrays.J.Korea Soc.Math.Educ.Ser.B Pure Appl.Math,13 (2006)311-318.
[13]G.-S.Cheon,and H.Kim.Simple proofs of open problems about the structure of involutions in the Riordan group.Linear Algebra and its Applications,428 (2008)930-940.
[14]G.-S.Cheon,H.Kim,and L.W.Shapiro.Riordan group involutions.Linear Algebra and its Applications,428 (2008)941-952.
[15]W.Y.C.Chen,N.Y.Li,L.W.Shapiro,and S.H.F.Yan.Matrix identities on weighted partial Motzkin paths.European J.Combinatorics,28 (2007)1196-1207.
[16]L.Comtet.Advanced Comibinomial.Rcidel,Dordrecht,NL,1974.
[17]C.Corsani,D.Merlini,and R.Sprugnoli.Left inversion of combinatorial sums.Discrete Mathematics,180 (1998)107-122.
[18]J.L.Diaz-Barrcro.J.Gibergans-Baguena,and P.G.Popescu.Some identities involving rational sums.Applicable Analysis and Discrete Mathematics.1 (2007)397-402.
[19]G.P.Egorycgev.Integral Representation and the Combinatorial Sums (trans.H.H.McFadden),volume 59.Amer.Math.Soc,Providence,1984.
[20]G.P.Egorychev,and E.V.Zima.Decomposition and group theoretic characterization of pairs of inverse relations of the Riordan type.Acta Applicandae Mathematicae,85 (2005)93-109.
[21]S.Getu,and L.W.Shapiro.Lattice paths and Bessel functions.Congressus Numerantium,108 (1995)161-169.
[22]H.W.Gould.Combinatorial Identitcs Morgantown:Morgarotion Printing and Binding Co,1972.
[23]R.L.Graham,D.E.Knuth,O.Patashnik.Concrete Mathematics reading Addison-Wesley,1988.
[24]T.X.He,L.C.Hsu,and P.J.-S.Shiue.The Sheffer group and the Riordan group.Discrete Applied Mathematics,155 (2007)1895-1909.
[25]D.S.Hough,and L.W.Shapiro.The noncrossing descent matrix is Riordan.Congressus Numerantium,162 (2003)83-96.
[26]I-C.Huang.Inverse relations and Schaudcr bases.J.Combin.Theory Scr.A,97 (2002)203-224.
[27]W.P.Johnson.,The curious history of Faa di Bruno' formula,American Mathematical Monthly 109(2002)217-234.
[28]S.G.Kettle.Families enumerated by the Schr(o|¨)der-Etherington sequence and a renewal array it generates,in:Combinatorial Mathematics X,Lecture Notes in Math,1036 (Springer:Berlin)244-274.
[29]W.Lang.On polynomials related to derivatives of the generating function of Catalan numbers.Fibonacci Quarterly,40 (2002)299-313.
[30]D.Li,and S.Shang.Several computing formulas for combinatorial sums.Appl.Math.J.Chinese Univ.Ser.B.17 (2002)119-124.
[31]X.R.Ma.A generalization of the Kummer identity and its application to Fibonacci-Lucas sequences.Fibonacci Quarterly,36 (1998)339-347.
[32]X.R.Ma.Inverse chains of the Riordan group and their applications to combinatorial sums.J.Math.Res.Exposition,19 (1999)445-451.
[33]D.Merlini.Proper generating trees and their internal path length.Discrete Applied Mathematics,156 (2008)627-646.
[34]D.Merlini,D.G.Rogers,R.Sprugnoli,and M.C.Verri.On some alternative characterizations of Riordan arrays.Canadian J.Mathematics,49 (1997)301-320.
[35]D.Merlini.and R.Sprugnoli.A Riordan array proof of a curious identity.Integers,2 (2002)A8.
[36]D.Merlini,and R.Sprugnoli.Playing with some identities of Andrews.J.of Integer Sequences,10 (2007).
[37]D.Merlini,R.Sprugnoli,and M.C.Verri.Algebraic and combinatorial properties of simple,coloured walks.Trees in Algebra and Programming-LNCS 787,(1994)218-233.
[38]D.Merlini,R.Sprugnoli,and M.C.Verri.A uniform model for the storage utilization of B-trec-like structures.Information Processing Letters,57 (1996)53-58.
[39]D.Merlini,R.Sprugnoli,and M.C.Verri.The tennis ball problem.J.Combinatorial Theory Ser.A,99 (2002)307-344.
[40]D.Merlini,R.Sprugnoli,and M.C.Verri.Waiting patterns for a printer.Discrete Applied Mathematics,144 (2004)359-373.
[41]D.Merlini,R.Sprugnoli,and M.C.Verri.The Akiyama-Tanigawa transformation.Integers,5 (2005)A5.
[42]D.Merlini,R.Sprugnoli,and M.C.Verri.The Cauchy numbers.Discrete Mathematics,306 (2006)1906-1920.
[43]D.Merlini,R.Sprugnoli,and M.C.Verri.Combinatorial inversions and implicit Riordan arrays.Electronic Notes on Discrete Mathematics-Combinatorics,(2006)103-110.
[44]D.Merlini.R.Sprugnoli,M.C.Verri.Lagrange Inversion:when and how.Acta Appl.Math,(2006)233-249.
[45]D.Merlini,R.Sprugnoli,and M.C.Verri.Combinatorial sums and implicit Riordan arrays.Discrete Mathematics,2008.
[46]D.Merlini,R.Sprugnoli,and M.C.Verri.The method of coefficients.Amer.Math.Monthly (to appear).
[47]D.Merlini,F.Uncini,and M.C.Verri.A unified approach to the study of general and palyndromic compositions.Integers,4 (2004)A23.
[48]D.Merlini,and M.C.Verri.Generating trees and proper Riordan arrays.Descrete Mathematics,218 (2000)167-183.
[49]S.C.Milne,Gaurav Bhatnagar.A characterization of inverse relations.Discrete Mathematics,193 (1998)235-245.
[50]E.Munarini.Enumeration of order ideals of a garland.Ars Combinatorica,76 (2005)185-192.
[51]A.Nkwanta.A Riordan matrix approach to unifying a selected class of combinatorial arrays.Congressus Numerantium,160 (2003)33-45.
[52]A.Nkwanta,and N.Knox.A note on Riordan matrices.African Americans in Mathematics,Ⅱ,(1999)99-107.
[53]A.Nkwanta,and L.W.Shapiro.Pell walks and Riordan matrices.Fibonacci Quarterly,43 (2005)170-180.
[54]P.Peart,and W.-J.Woan.Generating functions via Hankcl and Stieltjes matrices.J.of Integer Sequences,3 (2000).
[55]P.Peart,and W.-J.Woan.A divisibility property for a subgroup of Riordan matrices.Discrete Applied Mathematics,98 (2000)255-263.
[56]P.Peart,W.-J.Woan,and B.Tankcrslcy.Algebraic and combinatorial interpretations of the Genocchi triangle.Congressus Numerantium,175 (2005)45-51.
[57]P.Peart,and L.Woodson.Triple factorization of some Riordan matrices.Fibonacci Quarterly,31 (1993)121-128.
[58]J.Riordan.Combinatorial Identities.Wiley,New York.1968.
[59]D.G.Rogers.Pascal triangles,Catalan numbers and renewal arrays.Discrete Mathematics,22 (1978)301-310.
[60]S.Roman,The Umbral Calculus.Academic Press,New York,1984.
[61]L.W.Shapiro.A survey of the Riordan group.Talk at a meeting of the American Mathematical Society,Richmond,Virginia,1994.
[62]L.W.Shapiro.Some open questions about random walks,involutions,limiting distributions and generating functions.Advances in Applied Mathematics,27 (2001)585-596.
[63]L.W.Shapiro.Catalan trigonometry.Congressus Numerantium,156 (2002)129-136.
[64]L.W.Shapiro.Bijections and the Riordan group.Theoretical Computer Science,307(2003)403-413.
[65]L.W.Shapiro.The average is one.Congressus Numerantium,176 (2005)3-10.
[66]L.W.Shapiro,S.Getu,W.-J.Woan,and L.Woodson.The Riordan group.Discrete Applied Mathematics,34 (1991)229-239.
[67]R.Sprugnoli.Riordan arrays and combinatorial sums.Discrete Mathematics,132 (1994)267-290.
[68]R.Sprugnoli.Riordan arrays and the Abel-Gould identity.Discrete Mathematics,142 (1995)213-233.
[69]Y.-D.Sun,and C.Jia.Counting Dyck paths with strictly increasing peak sequences.J.Math.Res.Exposition,27(2007) 253-263.
[70]M.Tan,and T.Wang.Lab matrix and its algebraic properties.Ars Combinatorica,70(2004) 97-108.
[71]T.Wang,H.Yu,H.Yao.A matrix representation of combinatorial numbers with applications.Journal of Dalian University of Technology,(1996) 380-385.
[72]王天明.近代组合学.大连理工大学出版社,2008
[73]Herbert S.Wilf著,王天明译.发生函数论.清华大学出版社.2003.
[74]M.C.Wilson.Asymptotics for generalized Riordan arrays.2005 Int.Conf.on Analysis of Algorithms-Discrete Math.Thcor.Comput.Sci.Proc,(2005) 323-333.
[75]W.-J.Woan,and D.Hough.Lattice paths and subgroups of Riordan group.Congressus Numerantium,177(2005) 45-49.
[76]D.H.Yin.Riordan array/partial monoid.J.Math.Res.Exposition,23(2003) 253-260.
[77]D.S.Yin.Riordan groups and three generalized identities.J.Dalian Univ.TEch.,39(1999) 6-11.
[78]Q.-W.Zhang,and X.-R.Ma.The ordinary Bailey lemma and Riordan chain.J.Math.Res.Exposition,22(2002) 401-406.
[79]X.Zhao,and S.Ding.Sequences related to Riordan arrays.Fibonacci Quarterly,40(2002) 247-252.
[80]X.Zhao,S.Ding,and T.Wang.Some summation rules related to the Riordan arrays.Discrete Mathematics,281(2004) 295-307.
[81]X.Zhao,and T.Wang.Some identities related to reciprocal functions.Discrete Mathematics,265(2003) 323-335.
[82]X.Q.Zhao,Y.F.Zhang and A.W.Liang.A method of forming normal Riordan matrices.J.Luoyang Univ.,4(2001) 4-5.
[2]M.Aigner.A characterization of Bell numbers.Discrete Mathematics,205 (1999)207-210.
[3]George E.Andrews,Richard Askey,Ranjan Roy.Special Function,清华大学出版社．2004．
[4]D.Baccherini,D.Mcrlini,and R.Sprugnoli.Level generating trees and proper Riordan arrays.Applicable Analysis and Discrete Mathematics,2 (2008)69-91.
[5]P.Barry.A Catalan transform and related transformations on integer sequences.J.of Integer Sequences,8 (2005).
[6]P.Barry.On a family of generalized Pascal triangles denned by exponential Riordan arrays.J.of Integer Sequences,10 (2007).
[7]P.Barry.Some observations on the Lah and Laguerre transforms of integer sequences.J.of Integer Sequences,10 (2007).
[8]P.Barry.A note on Krawtchouk polynomials and Riordan arrays.J.of Integer Sequences,11 (2008).
[9]P.Barry,and P.Fitzpatrick.On a one-parameter family of Riordan arrays and the weight distribution of MDS codes.J.of Integer Sequences,9 (2007).
[10]N.T.Cameron,and A.Nkwanta.On some (pseudo)involutions in the Riordan group.J.of Integer Sequences,8 (2005).
[11]G.-S.Cheon,and M.E.A.El-Mikkawy.Generalized harmonic numbers with Riordan arrays.J.of Number Theory,128 (2008)413-425.
[12]G.-S.Cheon.M.E.A.El-Mikkawy,and H.-G.Seol.New identities for Stirling numbers via Riordan arrays.J.Korea Soc.Math.Educ.Ser.B Pure Appl.Math,13 (2006)311-318.
[13]G.-S.Cheon,and H.Kim.Simple proofs of open problems about the structure of involutions in the Riordan group.Linear Algebra and its Applications,428 (2008)930-940.
[14]G.-S.Cheon,H.Kim,and L.W.Shapiro.Riordan group involutions.Linear Algebra and its Applications,428 (2008)941-952.
[15]W.Y.C.Chen,N.Y.Li,L.W.Shapiro,and S.H.F.Yan.Matrix identities on weighted partial Motzkin paths.European J.Combinatorics,28 (2007)1196-1207.
[16]L.Comtet.Advanced Comibinomial.Rcidel,Dordrecht,NL,1974.
[17]C.Corsani,D.Merlini,and R.Sprugnoli.Left inversion of combinatorial sums.Discrete Mathematics,180 (1998)107-122.
[18]J.L.Diaz-Barrcro.J.Gibergans-Baguena,and P.G.Popescu.Some identities involving rational sums.Applicable Analysis and Discrete Mathematics.1 (2007)397-402.
[19]G.P.Egorycgev.Integral Representation and the Combinatorial Sums (trans.H.H.McFadden),volume 59.Amer.Math.Soc,Providence,1984.
[20]G.P.Egorychev,and E.V.Zima.Decomposition and group theoretic characterization of pairs of inverse relations of the Riordan type.Acta Applicandae Mathematicae,85 (2005)93-109.
[21]S.Getu,and L.W.Shapiro.Lattice paths and Bessel functions.Congressus Numerantium,108 (1995)161-169.
[22]H.W.Gould.Combinatorial Identitcs Morgantown:Morgarotion Printing and Binding Co,1972.
[23]R.L.Graham,D.E.Knuth,O.Patashnik.Concrete Mathematics reading Addison-Wesley,1988.
[24]T.X.He,L.C.Hsu,and P.J.-S.Shiue.The Sheffer group and the Riordan group.Discrete Applied Mathematics,155 (2007)1895-1909.
[25]D.S.Hough,and L.W.Shapiro.The noncrossing descent matrix is Riordan.Congressus Numerantium,162 (2003)83-96.
[26]I-C.Huang.Inverse relations and Schaudcr bases.J.Combin.Theory Scr.A,97 (2002)203-224.
[27]W.P.Johnson.,The curious history of Faa di Bruno' formula,American Mathematical Monthly 109(2002)217-234.
[28]S.G.Kettle.Families enumerated by the Schr(o|¨)der-Etherington sequence and a renewal array it generates,in:Combinatorial Mathematics X,Lecture Notes in Math,1036 (Springer:Berlin)244-274.
[29]W.Lang.On polynomials related to derivatives of the generating function of Catalan numbers.Fibonacci Quarterly,40 (2002)299-313.
[30]D.Li,and S.Shang.Several computing formulas for combinatorial sums.Appl.Math.J.Chinese Univ.Ser.B.17 (2002)119-124.
[31]X.R.Ma.A generalization of the Kummer identity and its application to Fibonacci-Lucas sequences.Fibonacci Quarterly,36 (1998)339-347.
[32]X.R.Ma.Inverse chains of the Riordan group and their applications to combinatorial sums.J.Math.Res.Exposition,19 (1999)445-451.
[33]D.Merlini.Proper generating trees and their internal path length.Discrete Applied Mathematics,156 (2008)627-646.
[34]D.Merlini,D.G.Rogers,R.Sprugnoli,and M.C.Verri.On some alternative characterizations of Riordan arrays.Canadian J.Mathematics,49 (1997)301-320.
[35]D.Merlini.and R.Sprugnoli.A Riordan array proof of a curious identity.Integers,2 (2002)A8.
[36]D.Merlini,and R.Sprugnoli.Playing with some identities of Andrews.J.of Integer Sequences,10 (2007).
[37]D.Merlini,R.Sprugnoli,and M.C.Verri.Algebraic and combinatorial properties of simple,coloured walks.Trees in Algebra and Programming-LNCS 787,(1994)218-233.
[38]D.Merlini,R.Sprugnoli,and M.C.Verri.A uniform model for the storage utilization of B-trec-like structures.Information Processing Letters,57 (1996)53-58.
[39]D.Merlini,R.Sprugnoli,and M.C.Verri.The tennis ball problem.J.Combinatorial Theory Ser.A,99 (2002)307-344.
[40]D.Merlini,R.Sprugnoli,and M.C.Verri.Waiting patterns for a printer.Discrete Applied Mathematics,144 (2004)359-373.
[41]D.Merlini,R.Sprugnoli,and M.C.Verri.The Akiyama-Tanigawa transformation.Integers,5 (2005)A5.
[42]D.Merlini,R.Sprugnoli,and M.C.Verri.The Cauchy numbers.Discrete Mathematics,306 (2006)1906-1920.
[43]D.Merlini,R.Sprugnoli,and M.C.Verri.Combinatorial inversions and implicit Riordan arrays.Electronic Notes on Discrete Mathematics-Combinatorics,(2006)103-110.
[44]D.Merlini.R.Sprugnoli,M.C.Verri.Lagrange Inversion:when and how.Acta Appl.Math,(2006)233-249.
[45]D.Merlini,R.Sprugnoli,and M.C.Verri.Combinatorial sums and implicit Riordan arrays.Discrete Mathematics,2008.
[46]D.Merlini,R.Sprugnoli,and M.C.Verri.The method of coefficients.Amer.Math.Monthly (to appear).
[47]D.Merlini,F.Uncini,and M.C.Verri.A unified approach to the study of general and palyndromic compositions.Integers,4 (2004)A23.
[48]D.Merlini,and M.C.Verri.Generating trees and proper Riordan arrays.Descrete Mathematics,218 (2000)167-183.
[49]S.C.Milne,Gaurav Bhatnagar.A characterization of inverse relations.Discrete Mathematics,193 (1998)235-245.
[50]E.Munarini.Enumeration of order ideals of a garland.Ars Combinatorica,76 (2005)185-192.
[51]A.Nkwanta.A Riordan matrix approach to unifying a selected class of combinatorial arrays.Congressus Numerantium,160 (2003)33-45.
[52]A.Nkwanta,and N.Knox.A note on Riordan matrices.African Americans in Mathematics,Ⅱ,(1999)99-107.
[53]A.Nkwanta,and L.W.Shapiro.Pell walks and Riordan matrices.Fibonacci Quarterly,43 (2005)170-180.
[54]P.Peart,and W.-J.Woan.Generating functions via Hankcl and Stieltjes matrices.J.of Integer Sequences,3 (2000).
[55]P.Peart,and W.-J.Woan.A divisibility property for a subgroup of Riordan matrices.Discrete Applied Mathematics,98 (2000)255-263.
[56]P.Peart,W.-J.Woan,and B.Tankcrslcy.Algebraic and combinatorial interpretations of the Genocchi triangle.Congressus Numerantium,175 (2005)45-51.
[57]P.Peart,and L.Woodson.Triple factorization of some Riordan matrices.Fibonacci Quarterly,31 (1993)121-128.
[58]J.Riordan.Combinatorial Identities.Wiley,New York.1968.
[59]D.G.Rogers.Pascal triangles,Catalan numbers and renewal arrays.Discrete Mathematics,22 (1978)301-310.
[60]S.Roman,The Umbral Calculus.Academic Press,New York,1984.
[61]L.W.Shapiro.A survey of the Riordan group.Talk at a meeting of the American Mathematical Society,Richmond,Virginia,1994.
[62]L.W.Shapiro.Some open questions about random walks,involutions,limiting distributions and generating functions.Advances in Applied Mathematics,27 (2001)585-596.
[63]L.W.Shapiro.Catalan trigonometry.Congressus Numerantium,156 (2002)129-136.
[64]L.W.Shapiro.Bijections and the Riordan group.Theoretical Computer Science,307(2003)403-413.
[65]L.W.Shapiro.The average is one.Congressus Numerantium,176 (2005)3-10.
[66]L.W.Shapiro,S.Getu,W.-J.Woan,and L.Woodson.The Riordan group.Discrete Applied Mathematics,34 (1991)229-239.
[67]R.Sprugnoli.Riordan arrays and combinatorial sums.Discrete Mathematics,132 (1994)267-290.
[68]R.Sprugnoli.Riordan arrays and the Abel-Gould identity.Discrete Mathematics,142 (1995)213-233.
[69]Y.-D.Sun,and C.Jia.Counting Dyck paths with strictly increasing peak sequences.J.Math.Res.Exposition,27(2007) 253-263.
[70]M.Tan,and T.Wang.Lab matrix and its algebraic properties.Ars Combinatorica,70(2004) 97-108.
[71]T.Wang,H.Yu,H.Yao.A matrix representation of combinatorial numbers with applications.Journal of Dalian University of Technology,(1996) 380-385.
[72]王天明.近代组合学.大连理工大学出版社,2008
[73]Herbert S.Wilf著,王天明译.发生函数论.清华大学出版社.2003.
[74]M.C.Wilson.Asymptotics for generalized Riordan arrays.2005 Int.Conf.on Analysis of Algorithms-Discrete Math.Thcor.Comput.Sci.Proc,(2005) 323-333.
[75]W.-J.Woan,and D.Hough.Lattice paths and subgroups of Riordan group.Congressus Numerantium,177(2005) 45-49.
[76]D.H.Yin.Riordan array/partial monoid.J.Math.Res.Exposition,23(2003) 253-260.
[77]D.S.Yin.Riordan groups and three generalized identities.J.Dalian Univ.TEch.,39(1999) 6-11.
[78]Q.-W.Zhang,and X.-R.Ma.The ordinary Bailey lemma and Riordan chain.J.Math.Res.Exposition,22(2002) 401-406.
[79]X.Zhao,and S.Ding.Sequences related to Riordan arrays.Fibonacci Quarterly,40(2002) 247-252.
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