By means of the modified Abel lemma on summation by parts, we examine a class of terminating balanced q-series. Two transformation formulae are established that contain ten summation formulae as consequences. | Turk J Math (2018) 42: 2699 – 2706 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Evaluating a class of balanced q -series 1 Wenchang CHU1,2,∗, School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou, . China 2 Department of Mathematics and Physics, University of Salento, Lecce, Italy Received: • Accepted/Published Online: • Final Version: Abstract: By means of the modified Abel lemma on summation by parts, we examine a class of terminating balanced q -series. Two transformation formulae are established that contain ten summation formulae as consequences. Key words: Abel’s lemma on summation by parts, basic hypergeometric series, terminating balanced series q -Pfaff– Saalschütz theorem 1. Introduction and motivation Let N be the set of natural numbers with N0 = {0} ∪ N . For an indeterminate x , the shifted factorial with the base q is defined by (x; q)0 = 1 and (x; q)n = (1 − x)(1 − qx) · · · (1 − q n−1 x) for n ∈ N. Its quotient form will be abbreviated as follows: [ ] (α; q)n (β; q)n · · · (γ; q)n α, β, · · · , γ . q = A, B, · · · , C (A; q)n (B; q)n · · · (C; q)n n This paper will investigate the following balanced series: [ n ] (λxy; q 3 ) ∑ x, y k k n Ωm (λ, x, y) = q , q qλ k (xy; q)2k (1) k=−m n (λ, x, y) ωm = [ n ∑ k=−m ] (qxy; q) λ 2k qk . q 3 λxy; q 3 ) qx, qy (q k k (2) By making the replacement k → −k on the summation index, we can check without difficulty that they satisfy the following reciprocal relation: Ωnm (λ, x, y) = ωnm (1/λ, 1/x, 1/y). In 1979, Andrews [1, Eq. ] (see also Gessel and Stanton [10, Eq. [ n [ −n n ] ∑ (y; q 3 )k k q, q 2 q ,q y q = χ(n ≡3 0) q qy, q 2 y q k (y; q)2k k=0 (3) ]) found the following identity: ] n 3 y⌊ 3 ⌋ , (4) q ⌊n 3⌋ ∗Correspondence: 2000 AMS Mathematics Subject Classification: Primary 33D15, Secondary .
