插值公式和插入算法在q-级数中的应用
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摘要
基本超几何级数,简称为q-级数,在过去二十多年发展极为迅速,并在组合学、数论、物理学和计算机代数学中有着广泛应用。多项式插值理论足逼近论和数值计算中重要的研究内容。利用微分算了这一工具,本文将深入研究插值公式在基本超几何级数中的应用(第二、三、四章)。同时,构造性的分拆理论也是一个非常丰富的学科,它包含许多经典的结论,这些结论在二十世纪深深影响了计数组合学的发展。本文将利用整数分拆的生成函数这一组合工具和插入算法的思想来证明基本超几何恒等式(第五章)。
第一章主要介绍基本超几何级数理论中的基本概念和背景,包括双边超几何级数,椭圆超几何级数。同时我们回顾了一些经典的组合恒等式。
第二章主要涉及一些已知的展开式,包括牛顿和拉格朗日插值公式,q-泰勒展开式,牛顿类型的有理插值公式,刘治国的展开式等,基于这些展开式我们得出一些重要的组合恒等式。
在第三章中,我们给出出一个2n点的插值公式,并深入研究了该插值公式在基本超几何级数中的应用。我们发现很多经典的q-级数恒等式都可以由该公式直接导出,包括终止的6φ和式,Dixon定理的q-模拟,五重积的有限形式,与除数函数有关的一些q-恒等式,双基超几何等式等。
在第四章中,我们定义了一个新的微分算子,由此给出了一个4n点的插值公式。随后,我们将该4n点的插值公式推广到椭圆超几何级数,给出了该插值公式的椭圆模拟。并且我们分别研究了该4n点的插值公式及其椭圆模拟在基本超几何级数和椭圆超几何级数中的应用。
在第五章中,我们主要利用插入算法的思想来证明基本超几何恒等式。Z-算法是由组合数学家Zeilberger提出的一种证明分拆恒等式的插入算法。在本章中,我们将利用该算法的思想给出一个新的插入算法,记作Z-算法的变形。最后,我们利用该变形给出基本超几何级数中最经典的等式—拉马努金恒等式
一个组合证明。
Basic hypergeometric series, shortened as q-series, has developed rapidly during the past two decades and has diverse applications in Analysis, Number Theory, Com-binatorics, Physics, and Computer Algebra. Interpolations, by polynomials or other functions, are fundamental subjects in Approximation Theory and Numerical Analysis. Moreover, Constructive Partition Theory is a rich subject, with many classical and im-portant results which influenced the development of Enumerative Combinatorics in the twentieth century.
The main results of this thesis consist of two approaches to basic hypergeometric series, including divided difference operators for interpolation formulas and insertion algorithm. This thesis is organized as follows.
The first chapter is devoted to an introduction to basic hypergeometric series, in-cluding bilateral hypergeometric series and elliptic hypergeometric series. We will give a brief history and basic definitions, as well as some classical q-identities.
In Chapter2, we shall give an overview of some well-known expansion formulas, including Newton and Lagrange interpolation formulas, q-Taylor formula, a Newton type rational interpolation formula,q-expansion formula of Liu and so on. We are also concerned with the applications of these expansion formulas, from which some important q-identities can be derived.
In Chapter3, we shall give a2n-point interpolation formula, which allows us to recover many important classical q-identities, such as the terminating6φ5summation formula, Bailey's q-analogues of Dixon's theorem, q-identities related to divisor func-tions, finite forms of the quintuple product identity, as well as a bibasic hypergeometric identity.
In Chapter4, we will give a new derivative operator, based on which a4n-point in-terpolation formula is derived. Then we generalize our result to elliptic hypergeometric series and give an elliptic analogue of the4n-point interpolation formula. Some appli-cations of these two interpolation formulas to basic hypergeometric series and elliptic hypergeometric series will also be discussed.
In Chapter5, we will focus on the applications of insertion algorithm in proving q-identities. We recall an insertion algorithm, raised by Zeilberger and named Algo-rithm Z. Following the idea of Algorithm Z, we give a new insertion algorithm, called the variation of Algorithm Z. In the end, by means of the variation, we give a new combinatorial proof of Ramanujan's1Ψ1summation formula, which is one of the most classical identities in q-series.
第一章主要介绍基本超几何级数理论中的基本概念和背景,包括双边超几何级数,椭圆超几何级数。同时我们回顾了一些经典的组合恒等式。
第二章主要涉及一些已知的展开式,包括牛顿和拉格朗日插值公式,q-泰勒展开式,牛顿类型的有理插值公式,刘治国的展开式等,基于这些展开式我们得出一些重要的组合恒等式。
在第三章中,我们给出出一个2n点的插值公式,并深入研究了该插值公式在基本超几何级数中的应用。我们发现很多经典的q-级数恒等式都可以由该公式直接导出,包括终止的6φ和式,Dixon定理的q-模拟,五重积的有限形式,与除数函数有关的一些q-恒等式,双基超几何等式等。
在第四章中,我们定义了一个新的微分算子,由此给出了一个4n点的插值公式。随后,我们将该4n点的插值公式推广到椭圆超几何级数,给出了该插值公式的椭圆模拟。并且我们分别研究了该4n点的插值公式及其椭圆模拟在基本超几何级数和椭圆超几何级数中的应用。
在第五章中,我们主要利用插入算法的思想来证明基本超几何恒等式。Z-算法是由组合数学家Zeilberger提出的一种证明分拆恒等式的插入算法。在本章中,我们将利用该算法的思想给出一个新的插入算法,记作Z-算法的变形。最后,我们利用该变形给出基本超几何级数中最经典的等式—拉马努金恒等式
一个组合证明。
Basic hypergeometric series, shortened as q-series, has developed rapidly during the past two decades and has diverse applications in Analysis, Number Theory, Com-binatorics, Physics, and Computer Algebra. Interpolations, by polynomials or other functions, are fundamental subjects in Approximation Theory and Numerical Analysis. Moreover, Constructive Partition Theory is a rich subject, with many classical and im-portant results which influenced the development of Enumerative Combinatorics in the twentieth century.
The main results of this thesis consist of two approaches to basic hypergeometric series, including divided difference operators for interpolation formulas and insertion algorithm. This thesis is organized as follows.
The first chapter is devoted to an introduction to basic hypergeometric series, in-cluding bilateral hypergeometric series and elliptic hypergeometric series. We will give a brief history and basic definitions, as well as some classical q-identities.
In Chapter2, we shall give an overview of some well-known expansion formulas, including Newton and Lagrange interpolation formulas, q-Taylor formula, a Newton type rational interpolation formula,q-expansion formula of Liu and so on. We are also concerned with the applications of these expansion formulas, from which some important q-identities can be derived.
In Chapter3, we shall give a2n-point interpolation formula, which allows us to recover many important classical q-identities, such as the terminating6φ5summation formula, Bailey's q-analogues of Dixon's theorem, q-identities related to divisor func-tions, finite forms of the quintuple product identity, as well as a bibasic hypergeometric identity.
In Chapter4, we will give a new derivative operator, based on which a4n-point in-terpolation formula is derived. Then we generalize our result to elliptic hypergeometric series and give an elliptic analogue of the4n-point interpolation formula. Some appli-cations of these two interpolation formulas to basic hypergeometric series and elliptic hypergeometric series will also be discussed.
In Chapter5, we will focus on the applications of insertion algorithm in proving q-identities. We recall an insertion algorithm, raised by Zeilberger and named Algo-rithm Z. Following the idea of Algorithm Z, we give a new insertion algorithm, called the variation of Algorithm Z. In the end, by means of the variation, we give a new combinatorial proof of Ramanujan's1Ψ1summation formula, which is one of the most classical identities in q-series.
引文
[1]G.E. Andrews, Enumerative proofs of certain q-identities, Glasgow Math. J.,8 (1967),33-40.
[2]G.E. Andrews, On Ramanujan's summation of 1Ψ1(a;b;q,z), Proc. Amer. Math. Soc.,22(1969),552-553.
[3]G.E. Andrews, The Theory of Partitions, Encyclopedia of Math, and its Applica-tions,2, Addison Wesley, Reading, (1976).
[4]G.E. Andrews, Number Theory, W.B. Saunders, Philadelphia; reprinted by Hin-dustan Publishing Co., New Delhi,1984.
[5]G.E. Andrews, q-Series:Their Development and Application in Analysis, Num-ber Theory, Combinatorics, Physics, and Computer Algebra, Regional Conf. Ser. in Math., no.66, Amer. Math. Soc., Providence, R.I., (1986).
[6]G.E. Andrews and R. Askey, A simple proof of Ramanujan's summation of 1Ψ1{a;b;q,z), Aequationes Math.,18 (1978),333-337.
[7]G.E. Andrews and D.M. Bressoud, Identities in combinatorics, III. Further aspects of ordered set sorting, Discrete Math.,49 (1984),223-236.
[8]G.E. Andrews and K. Uchimura, Identities in combinatorics, IV:differentiation and Harmonic numbers, Utilitas Math.,28 (1985),265-269.
[9]W.N. Bailey, On the analogue of Dixon's theorem of bilateral basic hypergeomet-ric series, Quart. J. Math.,1 (1950),318-320.
[10]W.N. Bailey, A note on certain q-identities, Quart. J. Math.,3 (1952),29-31.
[11]B.C. Berndt, Number Theory in the Spirit of Ramanujan, Amer. Math. Soc., Prov-idence, Rohode Island,2006.
[12]B.C. Berndt, B. Kim, A.J. Yee, Ramanujan's lost notebook:Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Combin. Theory Ser. A,117 (2010),957-973.
[13]C. Bessenrodt, On a theorem of Alladi and Gordon and the Gaussian polynomials, J. Combin. Theory Ser. A,69 (1995),159-167.
[14]D.M. Bressoud and D. Zeilberger, Generalized Rogers-Ramanujan bijections, Ad-v. Math.,78 (1989),42-75.
[15]L. Carlitz, Some q-expansion formulas, Glas. Mat.,8 (28) (1973),205-214.
[16]A.L. Cauchy, Memoire sur les Fonctions dont plusiers valeurs, Comp. Rend. Acad Sci. Paris,17 (1843),567-572.
[17]S.H.L. Chen, W.Y.C. Chen, A.M. Fu and W.J.T Zang, The Algorithm Z and Ra-manujan's 1Ψ1 summation, Ramanujan J.,25 (2011),37-47.
[18]S.H.L. Chen and A.M. Fu, A 2n-point interpolation formula with its applications to q-identities, Discrete Math.,311 (2011),1793-1802.
[19]S.H.L. Chen and A.M. Fu, A 4n-point interpolation formula and its elliptic ana-logue, submitted.
[20]W.Y.C. Chen, A.M. Fu and B.Y. Zhang, The homogenoua q-difference operator, Adv. Appl. Math.,31 (2003),659-668.
[21]W.C. Chu, q-derivative operator and basic hypergeometric series, Results in Math.,49,25-44.
[22]T. Clausen, Ueber die Falle, wenn die Reihe von der Form ein Quadrat von der Form hat, J. reine angew. Math.,3,89-91.
[23]S. Corteel, Particle seas and basic hypergeometric series, Adv. Appl. Math.,31 (2003),199-214.
[24]S. Corteel and J. Lovejoy, Frobenius partitions and the combinatorics of Ramanu-jan's 1Ψ1 summation, J. Combin. Theory Ser. A,97 (2002),177-183.
[25]S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc.,356 (2004), 1623-1635.
[26]J.A. Daum, The basic analogue of Kummer's theorem, Bull. Amer. Math. Soc., 48 (1942),711-713.
[27]J. Dougall, On Vandermonde's theorem and some more general expansions, Proc. Edin. Math. Soc.,25(1907),114-132.
[28]L. Euler, Observations analyticae variae de combinationibus, Comm. Acad. Petrop.,13 (1751),64-93.
[29]N.J. Fine, Basic Hypergeometric Series and Its Applications, Amer. Math. Soc., (1988).
[30]A.M. Fu, A combinatorial proof of the Lebesgue identity, Discrete Math.,308 (2007),2611-2613.
[31]A.M. Fu and A. Lascoux,q-identities from Lagrange and Newton interpolation, Adv. Appl. Math.,31 (2003),527-531.
[32]A.M. Fu and A. Lascoux,q-identities related to overpartitions and divisor func-tions. Electron. J. Combin.,12 (2005), R38.
[33]A.M. Fu and A. Lascoux, A new type rational interpolation formula, Adv. Appl. Math.,41 (2008),452-458.
[34]A.M. Fu and A. Lascoux, Non-symmetric Cauchy kernels for the classical groups, J. Combin. Theory Ser. A,116 (2009),903-917.
[35]G. Gasper and M. Rahman, Basic Hypergeometric Series (second ed.), Encyclo-pedia Math. Appl., Vol.96, Cambridge Univ. Press, Cambridge,2004.
[36]V.J.W. Guo and J. Zeng, Short proofs of summation and transformation formulas for basic hypergeometric series, J. Math. Anal. Appl.,327 (2007),310-325.
[37]G.H. Hardy, Ramanujan, Cambridge University Press, Cambridge; reprinted by Chelsea, New York,1978.
[38]E. Heine, Untersuchungen Uber die Reihe, J. Reine angew. Math.,32 (1846), 210-212.
[39]E. Heine, Untersuchungen Uber die Reihe, J. Reine angew. Math.,34 (1847), 285-328.
[40]E. Heine, Handbuch der Kugelfunctionen, Theorie und Anwendungen, Reimer, Berlin.,1 (1878).
[41]M. Ismail, A simple proof of Ramanujan's summation of 1Ψ1(a;b;q,z), Proc. Amer. Math. Soc.,63 (1977),185-186.
[42]F.H. Jackson, Summation of q-hypergeometric series, Messenger Math.,50 (1921),101-112.
[43]J.T. Joichi and D. Stanton, Bijective proofs of basic hypergeometric series identi-ties. Pacific J. Math.,127 (1987),103-120.
[44]V. Kac and P. Cheung, Quantum Calculus, Springer, (2001).
[45]P. Kirchenhofer, A note on alternating sums, Electronic J. Combin.,3 (1996) R7,
[46]C. Krattenthaler, Generating functions for plane partitions of a given shape, Manuscripta Math.,69 (1990),173-201.
[47]C. Krattenthaler, The major counting of nonintersecting lattice paths and generat-ing functions for tableaux, Mem. Amer. Math. Soc.,552 (1995).
[48]E.E. Kummer, Uber die hypergeometrische Reihe, J. fur Math.,15 (1836),39-83 and 127-172.
[49]A. Lascoux, Symmetric Functions and Combinatorial Operators on Polynomials, CBMS/AMS Lecture Notes,99 (2003).
[50]Z.G. Liu, An expansion formula for q-series and applications, The Ramanujan J., 6:4 (2002),429-447.
[51]K. Mimachi, A proof of Ramanujan's identity by use of loop integral, SIAM J. Math. Anal.,19 (1988),1490-1493.
[52]I. Pak, Partition bijections, a survey, Ramanujan J.,12 (2006),5-75.
[53]H. Prodinger, Some applications of the q-Rice formula, Random Structures and Algorithms,19 (2001),552-557.
[54]S. Roman, More on the q-umbral calculus, J. Math. Anal. Appl.,107 (1985),222-252.
[55]M. Schlosser, Summation theorems for multidimensional basic hypergeometric series by determinant evaluations, Discrete Math.,210 (2000),151-169.
[56]V.P. Spiridonov, Theta hypergeometric series. In asymptotic combinatorics with application to mathematical physics (St. Petersburg,2001), volume 77 of NATO Sci. Ser. II Math. Phys. Chem., pages 307-327. Kluwer Acad. Publ., Dordrecht, 2002.
[57]V.P. Spiridonov, An elliptic incarnation of the Bailey chain, Int. Math. Res. Not., 37 (2002),1945-1977.
[58]K. Uchimura, An identity for the divisor generating function arising from sorting theory, J. Combin. Theory Ser. A,31 (1981),131-135.
[59]K. Uchimura, A generalization of identities for the divisor generating function, Utilitas Math.,25 (1984),377-379.
[60]S.O. Warnaar, Summation and transformation formulas for elliptic hypergeomet-ric series, Constr. Approx.,18 (2002),479-502.
[61]G.N. Watson, A new proof of the Rogers-Ramanujan identities, J. London Math. Soc.,4 (1929),4-9.
[62]A.J. Yee, Combinatorial proofs of Ramanujan's 1Ψ1 summation and the q-Gauss summation, J. Combin. Theory Ser. A,105 (2004),63-77.
[63]Z.Z. Zhang and J. Wang, Two operator identities and their applications to ter-minating basic hypergeometric series and q-integrals, J. Math. Anal. Appl.,312 (2005),653-665.
[2]G.E. Andrews, On Ramanujan's summation of 1Ψ1(a;b;q,z), Proc. Amer. Math. Soc.,22(1969),552-553.
[3]G.E. Andrews, The Theory of Partitions, Encyclopedia of Math, and its Applica-tions,2, Addison Wesley, Reading, (1976).
[4]G.E. Andrews, Number Theory, W.B. Saunders, Philadelphia; reprinted by Hin-dustan Publishing Co., New Delhi,1984.
[5]G.E. Andrews, q-Series:Their Development and Application in Analysis, Num-ber Theory, Combinatorics, Physics, and Computer Algebra, Regional Conf. Ser. in Math., no.66, Amer. Math. Soc., Providence, R.I., (1986).
[6]G.E. Andrews and R. Askey, A simple proof of Ramanujan's summation of 1Ψ1{a;b;q,z), Aequationes Math.,18 (1978),333-337.
[7]G.E. Andrews and D.M. Bressoud, Identities in combinatorics, III. Further aspects of ordered set sorting, Discrete Math.,49 (1984),223-236.
[8]G.E. Andrews and K. Uchimura, Identities in combinatorics, IV:differentiation and Harmonic numbers, Utilitas Math.,28 (1985),265-269.
[9]W.N. Bailey, On the analogue of Dixon's theorem of bilateral basic hypergeomet-ric series, Quart. J. Math.,1 (1950),318-320.
[10]W.N. Bailey, A note on certain q-identities, Quart. J. Math.,3 (1952),29-31.
[11]B.C. Berndt, Number Theory in the Spirit of Ramanujan, Amer. Math. Soc., Prov-idence, Rohode Island,2006.
[12]B.C. Berndt, B. Kim, A.J. Yee, Ramanujan's lost notebook:Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Combin. Theory Ser. A,117 (2010),957-973.
[13]C. Bessenrodt, On a theorem of Alladi and Gordon and the Gaussian polynomials, J. Combin. Theory Ser. A,69 (1995),159-167.
[14]D.M. Bressoud and D. Zeilberger, Generalized Rogers-Ramanujan bijections, Ad-v. Math.,78 (1989),42-75.
[15]L. Carlitz, Some q-expansion formulas, Glas. Mat.,8 (28) (1973),205-214.
[16]A.L. Cauchy, Memoire sur les Fonctions dont plusiers valeurs, Comp. Rend. Acad Sci. Paris,17 (1843),567-572.
[17]S.H.L. Chen, W.Y.C. Chen, A.M. Fu and W.J.T Zang, The Algorithm Z and Ra-manujan's 1Ψ1 summation, Ramanujan J.,25 (2011),37-47.
[18]S.H.L. Chen and A.M. Fu, A 2n-point interpolation formula with its applications to q-identities, Discrete Math.,311 (2011),1793-1802.
[19]S.H.L. Chen and A.M. Fu, A 4n-point interpolation formula and its elliptic ana-logue, submitted.
[20]W.Y.C. Chen, A.M. Fu and B.Y. Zhang, The homogenoua q-difference operator, Adv. Appl. Math.,31 (2003),659-668.
[21]W.C. Chu, q-derivative operator and basic hypergeometric series, Results in Math.,49,25-44.
[22]T. Clausen, Ueber die Falle, wenn die Reihe von der Form ein Quadrat von der Form hat, J. reine angew. Math.,3,89-91.
[23]S. Corteel, Particle seas and basic hypergeometric series, Adv. Appl. Math.,31 (2003),199-214.
[24]S. Corteel and J. Lovejoy, Frobenius partitions and the combinatorics of Ramanu-jan's 1Ψ1 summation, J. Combin. Theory Ser. A,97 (2002),177-183.
[25]S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc.,356 (2004), 1623-1635.
[26]J.A. Daum, The basic analogue of Kummer's theorem, Bull. Amer. Math. Soc., 48 (1942),711-713.
[27]J. Dougall, On Vandermonde's theorem and some more general expansions, Proc. Edin. Math. Soc.,25(1907),114-132.
[28]L. Euler, Observations analyticae variae de combinationibus, Comm. Acad. Petrop.,13 (1751),64-93.
[29]N.J. Fine, Basic Hypergeometric Series and Its Applications, Amer. Math. Soc., (1988).
[30]A.M. Fu, A combinatorial proof of the Lebesgue identity, Discrete Math.,308 (2007),2611-2613.
[31]A.M. Fu and A. Lascoux,q-identities from Lagrange and Newton interpolation, Adv. Appl. Math.,31 (2003),527-531.
[32]A.M. Fu and A. Lascoux,q-identities related to overpartitions and divisor func-tions. Electron. J. Combin.,12 (2005), R38.
[33]A.M. Fu and A. Lascoux, A new type rational interpolation formula, Adv. Appl. Math.,41 (2008),452-458.
[34]A.M. Fu and A. Lascoux, Non-symmetric Cauchy kernels for the classical groups, J. Combin. Theory Ser. A,116 (2009),903-917.
[35]G. Gasper and M. Rahman, Basic Hypergeometric Series (second ed.), Encyclo-pedia Math. Appl., Vol.96, Cambridge Univ. Press, Cambridge,2004.
[36]V.J.W. Guo and J. Zeng, Short proofs of summation and transformation formulas for basic hypergeometric series, J. Math. Anal. Appl.,327 (2007),310-325.
[37]G.H. Hardy, Ramanujan, Cambridge University Press, Cambridge; reprinted by Chelsea, New York,1978.
[38]E. Heine, Untersuchungen Uber die Reihe, J. Reine angew. Math.,32 (1846), 210-212.
[39]E. Heine, Untersuchungen Uber die Reihe, J. Reine angew. Math.,34 (1847), 285-328.
[40]E. Heine, Handbuch der Kugelfunctionen, Theorie und Anwendungen, Reimer, Berlin.,1 (1878).
[41]M. Ismail, A simple proof of Ramanujan's summation of 1Ψ1(a;b;q,z), Proc. Amer. Math. Soc.,63 (1977),185-186.
[42]F.H. Jackson, Summation of q-hypergeometric series, Messenger Math.,50 (1921),101-112.
[43]J.T. Joichi and D. Stanton, Bijective proofs of basic hypergeometric series identi-ties. Pacific J. Math.,127 (1987),103-120.
[44]V. Kac and P. Cheung, Quantum Calculus, Springer, (2001).
[45]P. Kirchenhofer, A note on alternating sums, Electronic J. Combin.,3 (1996) R7,
[46]C. Krattenthaler, Generating functions for plane partitions of a given shape, Manuscripta Math.,69 (1990),173-201.
[47]C. Krattenthaler, The major counting of nonintersecting lattice paths and generat-ing functions for tableaux, Mem. Amer. Math. Soc.,552 (1995).
[48]E.E. Kummer, Uber die hypergeometrische Reihe, J. fur Math.,15 (1836),39-83 and 127-172.
[49]A. Lascoux, Symmetric Functions and Combinatorial Operators on Polynomials, CBMS/AMS Lecture Notes,99 (2003).
[50]Z.G. Liu, An expansion formula for q-series and applications, The Ramanujan J., 6:4 (2002),429-447.
[51]K. Mimachi, A proof of Ramanujan's identity by use of loop integral, SIAM J. Math. Anal.,19 (1988),1490-1493.
[52]I. Pak, Partition bijections, a survey, Ramanujan J.,12 (2006),5-75.
[53]H. Prodinger, Some applications of the q-Rice formula, Random Structures and Algorithms,19 (2001),552-557.
[54]S. Roman, More on the q-umbral calculus, J. Math. Anal. Appl.,107 (1985),222-252.
[55]M. Schlosser, Summation theorems for multidimensional basic hypergeometric series by determinant evaluations, Discrete Math.,210 (2000),151-169.
[56]V.P. Spiridonov, Theta hypergeometric series. In asymptotic combinatorics with application to mathematical physics (St. Petersburg,2001), volume 77 of NATO Sci. Ser. II Math. Phys. Chem., pages 307-327. Kluwer Acad. Publ., Dordrecht, 2002.
[57]V.P. Spiridonov, An elliptic incarnation of the Bailey chain, Int. Math. Res. Not., 37 (2002),1945-1977.
[58]K. Uchimura, An identity for the divisor generating function arising from sorting theory, J. Combin. Theory Ser. A,31 (1981),131-135.
[59]K. Uchimura, A generalization of identities for the divisor generating function, Utilitas Math.,25 (1984),377-379.
[60]S.O. Warnaar, Summation and transformation formulas for elliptic hypergeomet-ric series, Constr. Approx.,18 (2002),479-502.
[61]G.N. Watson, A new proof of the Rogers-Ramanujan identities, J. London Math. Soc.,4 (1929),4-9.
[62]A.J. Yee, Combinatorial proofs of Ramanujan's 1Ψ1 summation and the q-Gauss summation, J. Combin. Theory Ser. A,105 (2004),63-77.
[63]Z.Z. Zhang and J. Wang, Two operator identities and their applications to ter-minating basic hypergeometric series and q-integrals, J. Math. Anal. Appl.,312 (2005),653-665.