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Abel’s lemma on summation by parts and Ramanujan’s \(_1\psi_1\)-series identity. (English) Zbl 1116.33018

The author shows how one can deduce Ramanujan’s \(_1 \psi _1\) summation from Abel’s lemma on summation by parts and Jacobi’s triple product idenitity. Specifically, applying Abel’s lemma in two different ways to the sum \(_1\psi_1(a,c;z)\) shows that this sum is the quotient \((c/a)_{m+n}/(c,c/az)_m(z,q/a)_n\) times \(_1\psi_1(a/q^n,cq^m;zq^n)\). Letting \(m,n \to \infty\) and using the triple product identity yields Ramanujan’s formula.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
05A30 \(q\)-calculus and related topics
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