Yet another proof of Ramanujan’s \(_1\psi_1\) sum. (English) Zbl 1515.33016
Summary: We give a simple proof of Ramanujan’s \(_1\psi_1\) sum. The present argument depends on the residue theorem and the \(q\)-binomial theorem only. Even the manipulation of the series is not used. We give also a proof of Bailey’s \(_6\psi_6\) sum in the same manner.
MSC:
| 33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |
References:
| [1] | Gasper, George, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, xxvi+428 pp. (2004), Cambridge University Press, Cambridge · Zbl 1129.33005 · doi:10.1017/CBO9780511526251 |
| [2] | Mimachi, Katsuhisa, A proof of Ramanujan’s identity by use of loop integrals, SIAM J. Math. Anal., 1490-1493 (1988) · Zbl 0665.33002 · doi:10.1137/0519112 |
| [3] | S. O. Warnaar, Ramanujan’s \(_1\psi_1\) summation, Notices Amer. Math. Soc. 60 (2013), no. 1, 18–22. |
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