n1
Fn (a, b, d) :=
k=0
1 q 5k a
b, d q 3 a/b2 d2
q
k
[qa/bd; q]3k b2 d2/q 2 4 4 a/b, q 4 a/d q [bd, bd/q, qbd; q 2 ]k q
qk ;
k
2
n1
W. Chu and C. Wang Gn (a, c, e) :=
k=0 n1
1 q 5k a 1q 5k a
k=0 n1
c2 e2/q 2 a3 q qa/c, qa/e
k
[qa2/ce, q 2 a2/ce, q 3 a2/ce; q 2 ]k c, e 4 6 a4/c2 e2 q q (ce/a; q)3k
qk ;
k
Un (a, b, d) :=
3 2 6 k (q 2 a/bd; q)k b, d b2 d2 /q 3 a 2 (q a /bd; q )k 3 q (2) ; 5 a2 /b2 d2 q 3 a/b, q 3 a/d q 3 k q (bd; q 2 )2k q k k (bd/q a)
Vn (a, c, e) :=
k=0
1 q 5k a
(a2 /ce; q 2 )2k qc2 e2 /a2 2 q (qce/a; q)k qa/c, qa/e
k
(a/ce)k qc, qe 3 2 a3 /c2 e2 q 5 ce; q 6 ) (q k q
q ( 2 ) .
k
k
By means of the series rearrangement, Gasper and Rahman [10, 11, 12] discovered several summation and transformation formulae for the nonterminating special cases of Fn (a, b, d) and Un (a, b, d). The terminating series identities for the last four sums have been established by Chu [4] and Chu–Wang [8], respectively, through inversion techniques and Abel's lemma on summation by parts. For the partial sums Fn (a, b, d) and Gn (a, c, e), the present authors [7] recently derived several useful reciprocal relations and transformation formulae in terms of wellpoised series. The purpose of this paper is to investigate the remaining two partial sums Un (a, b, d) and Vn (a, c, e). By utilizing the modified Abel lemma on summation by parts, we shall show six unusual transformation formulae with two between them and other four expressing Un (a, b, d) and Vn (a, c, e) as partial sums of quadratic and cubic series. Several new and known terminating as well as nonterminating series identities are consequently obtained as particular instances. In order to make the paper self-contained, we reproduce Abel's lemma on summation by parts (cf. [5, 6, 7, 8]) as follows. For an arbitrary complex sequence {τk }, define the backward and forward difference operators and , respectively, by τk = τk τk1 and τk = τk+1 τk .

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