Abel's Lemma On Summation By Parts And Q-Series Transformation And Summation Formulae

By means of Abel's lemma on summation by parts,this thesis investigates systematically the partial sums of basic hypergeometric series.Several transformation and summation formulae for well-poised,quadratic,cubic,quartic and other series are established.As preliminaries,the author briefly reviews the theory and development of basic hypergeometric series.The modified Abel lemma on summation by parts is proved.Its application to basic hypergeometric series is illustrated with well-poised series.In Chapter 2,Abel's lemma on summation by parts is applied to quadratic series. A reciprocal relation and a transformation to well-poised series for quadratic series are established.They are further explored to derive some interesting known identities due to Chu(1995),Gasper(1989),Gessel-Stanton(1983) and Rahman(1993),which have originally been discovered through inversion technique and series rearrangements.In Chapter 3,the modified Abel lemma is employed to investigate cubic series.A reciprocal relation for cubic series as well as a transformation from cubic series to wellpoised series are established,which not only generalize some results due to Chu(1993), Gasper(1989) and Gasper-Rahman(1983) but also lead to several new q-series identities.In Chapter 4,two pairs of dual quartic series are treated by means of Abel's lemma on summation by parts.Similar to the quadratic and cubic series,reciprocal relations and transformations to well-poised series are derived for the first pair.Instead for the second pair,six unusual transformation formulae are obtained with two between them and other four expressing each as partial sums of quadratic and cubic series.Finally,in the last chapter,by means of the nonterminating form of Abel's lemma on summation by parts,a common extension is accomplished for Andrews'(1973) q-Bailey and q-Gauss formulae as well as the q-analogues of Watson and Whipple formulae due to Andrews(1976) and Jain(1985),which can also be considered as a full q-analogue of M. Jackson's(1949) ₃H₃-series identity.