×

A note on a generalization of a q-series transformation of Ramanujan. (English) Zbl 0598.33004

An interesting generalization of a q-series transformation of Srinivasava Ramanujan (1887-1920) was given by S. Bhargava and C. Adiga [Indian J. Pure Appl. Math. 17, 338-342 (1986)]. In the present paper it is shown how readily this general q-series transformation would follow as a limiting case of Heine’s transformation for basic hypergeometric series. Several other interesting consequences of Heine’s result, relevant to the present discussion, are also deduced.

MSC:

33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
11P81 Elementary theory of partitions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] C. Adiga, B. C. Berndt, S. Bhargava and G. N. Watson: Chapter 16 of Ramanujan’s Second Notebook; Theta-functions and g-series, Mem. Amer. Math. Soc, vol. 53 (no. 315), Amer. Math. Soc, Providence, Rhode Island (1985). · Zbl 0565.33002
[2] G. E. Andrews: An introduction to Ramanujan’s ”lost” notebook. Amer. Math. Monthly, 86, 89-108 (1979). JSTOR: · Zbl 0401.01003 · doi:10.2307/2321943
[3] S. Bhargava and C. Adiga: On some continued fraction identities of Srinivasa Ramanujan. Proc. Amer. Math. Soc, 92, 13-18 (1984). JSTOR: · Zbl 0519.10006 · doi:10.2307/2045144
[4] ——; a basic hypergeometric transformation of Ramanujan and a generalization. Indian J. Pure Appl. Math., 17, 338-342 (1986). · Zbl 0596.33005
[5] L. Carlitz: Advanced problem no. 5196. Amer. Math. Monthly, 71, 440-441 (1964).
[6] L. Carlitz: Multiple sum-product identities, ibid., 72, 917-918 (1965).
[7] E. Heine: Untersuchungen uber die Reihe.... J. Reine Angew. Math., 34, 285-328 (1847). · ERAM 034.0971cj
[8] V. Ramamani and K. Venkatachaliengar: On a partition theory of Sylvester. Michigan Math. J., 19, 137-140 (1972). · Zbl 0236.10007 · doi:10.1307/mmj/1029000844
[9] S. Ramanujan: Notebooks of Srinivasa Ramanujan. Vols. I and II, Tata Institute of Fundamental Research, Bombay (1957). · Zbl 0138.24201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.