Abstract
The Abel method on summation by parts is reformulated to present new and elementary proofs of several classical identities of terminating balanced basic hypergeometric series. The examples strengthen our conviction that as traditional analytical instrument, the revised Abel method on summation by parts is indeed a very natural choice for working with basic hypergeometric series.
Highlights
For an arbitrary complex sequence {τk}, define the backward and forward difference operators ∇ and ·, respectively, by (1)∇τk = τk − τk−1 and · τk = τk − τk+1 where · is adopted for convenience in the present paper, which differs from the usual operator ∆ only in the minus sign. Abel’s lemma on summation by parts may be reformulated as ∞Bk∇Ak = [AB]∞ − A−1B0 + Ak · Bk k=0 k=0 provided that the limit [AB]∞ := lim m→∞AmBm+1 exists and one of the nonterminating series just displayed is convergent.according to the definition of the backward difference, we have m m Bk∇Ak =Bk Ak − Ak−1 k=0 k=0 m
If we let b = q−δ−2m with δ = 0, 1 in Theorem 5, we will obtain another terminating q-analogue of the above mentioned Watson’s summation formula
Abel’s lemma on summation by parts can be used to manipulate the M-series defined in the last section as follows:
Summary
Abel’s lemma on summation by parts, basic hypergeometric series. Applying Abel’s lemma on summation by parts to a given q-series Ω, the machinery establishes a recurrence relation. The limiting case m → ∞ of the transformation (if exists, ) leads to a nonterminating series identity.
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