A nonterminating balanced $q$-series is examined by means of the modified Abel lemma on summation by parts that leads to $q$-analogues of five difficult identities for classical hypergeometric series, including three formulae conjectured by Gosper in 1977.
Abel's lemma on summation by parts, classical hyper geometric series, basic hyper geometric series, balanced series, bilateral series, the gamma function
CHEN, XIAOJING and CHU, WENCHANG
"$Q$-analogues of five difficult hypergeometric evaluations,"
Turkish Journal of Mathematics: Vol. 44:
3, Article 12.
Available at: https://journals.tubitak.gov.tr/math/vol44/iss3/12