Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 5 (2009), 050, 19 pages
Partial Sums of Two Quartic q-Series
Wenchang CHU

and Chenying WANG
Dipartimento di Matematica, Universit` degli Studi di Salento, a Lecce-Arnesano P. O. Box 193, Lecce 73100, Italy E-mail: chu.wenchang@unile.it College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, P. R. China E-mail: wang.chenying@163.com
Received January 20, 2009, in final form April 17, 2009; Published online April 22, 2009 doi:10.3842/SIGMA.2009.050 Abstract. The partial sums of two quartic basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several summation and transformation formulae are consequently established. Key words: basic hypergeometric series (q-series); well-poised q-series; quadratic q-series; cubic q-series; quartic q-series; the modified Abel lemma on summation by parts 2000 Mathematics Subject Classication: 33D15; 05A15
1
Introduction and motivation
(x; q)0 = 1
∞
For two indeterminate x and q, the shifted-factorial of x with base q is defined by and (x; q)n = (1 x)(1 xq) 1 xq n1 for n ∈ N.
When |q| < 1, we have two well-defined infinite products (x; q)∞ =
k=0
1 qk x
and
(x; q)n = (x; q)∞ / xq n ; q
∞
.
The product and fraction of shifted factorials are abbreviated respectively to [α, β, . . . , γ; q]n = (α; q)n (β; q)n (γ; q)n , α, β, . . . , γ q A, B, . . . , C =
n
(α; q)n (β; q)n (γ; q)n . (A; q)n (B; q)n (C; q)n
∞ n=0
Following Gasper–Rahman [12], the basic hypergeometric series is defined by
1+r φs
a0 , a1 , . . . , ar q; z = b1 , . . . , b s
(1)n q ( 2 )
n
sr
a0 , a1 , . . . , ar q q, b1 , . . . , bs
zn,
n
where the base q will be restricted to |q| < 1 for nonterminating q-series. For its connections to special functions and orthogonal polynomials, the reader can refer, for example, to the monograph written by Andrews–Askey–Roy [2] and the paper by Koornwinder [15]. In the theory of basic hypergeometric series, there are several important classes, for example, well-poised [3], quadratic [13, 17], cubic [16] and quartic [10, 11] series. To our knowledge, there are four quartic series which can be displayed as follows:

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