Then Abel's lemma on summation by parts can be modified as follows:
n1 n1
Bk
k=0
Ak = An1 Bn A1 B0
k=0
Ak Bk .
This can be considered as the discrete counterpart for the integral formula
b b
f (x)g (x)dx = f (b)g(b) f (a)g(a)
a a
f (x)g(x)dx.
In fact, it is almost trivial to check the following expression
n1 n1 n1 n1
Bk
k=0
Ak =
k=0
Bk Ak Ak1 =
k=0
Ak B k
k=0
Ak1 Bk .
Replacing k by k + 1 for the last sum, we can reformulate the equation as follows
n1 n1
Bk
k=0
Ak = An1 Bn A1 B0 +
k=0 n1
Ak Bk Bk+1
= An1 Bn A1 B0
k=0
Ak Bk ,
which is exactly the equality stated in the modified Abel lemma.
Partial Sums of Two Quartic q-Series
3
Throughout the paper, if Wn is used to denote the partial sum of some q-series, then the corresponding letter W without subscript will stand for the limit of Wn (if it exists of course) when n → ∞. When applying the modified Abel lemma on summation by parts to deal with hypergeometric series, the crucial step lies in finding shifted factorial fractions {Ak , Bk } so that their differences are expressible as ratios of linear factors. This has not been an easy task, even though it is indeed routine to factorize {Ak , Bk } once they are figured out. Specifically for Un (a, b, d) and Vn (a, b, d), we shall devise three well-poised difference pairs for each partial sum. This is based on numerous attempts to detect Ak and Bk sequences such that not only their differences turn out to be factorizable, but also their combinations match exactly the summands displayed in Un (a, b, d) and Vn (a, b, d). The contents of the paper will be organized as follows. In the second section, Un (a, b, d) will be reformulated through the modified Abel lemma on summation by parts, which lead to three transformations of Un (a, b, d) into partial sums of quadratic, cubic and quartic series. Then the third section will be devoted to the transformation formulae of Vn (a, b, d) in terms of partial sums of quadratic, cubic and quartic series. These transformations on Un (a, b, d) and Vn (a, b, d) will recover several known identities appeared in Chu–Wang [4, 8] and Gasper–Rahman [12], and yield a few additional new summation formulae.

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