Novel results on transformation identities for bilateral basic hypergeometric series
Ranu Singh2024104857.ranu@dr.sharda.ac.inDepartment of Mathematics & Data ScienceSharda School of Engineering & ScienceSharda UniversityGreater Noida, Uttar Pradesh, 201310, IndiaView full profile → , *Nidhi SahniCorresponding authornidhi.sahni@sharda.ac.inDepartment of Mathematics & Data ScienceSharda School of Engineering & ScienceSharda UniversityGreater Noida, Uttar Pradesh, 201310, IndiaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Jan 2026
- Published Online:
- 30 Sep 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2706
- Pages:
- 2749–2757
Abstract
The present work develops a systematic approach to construct some transformation formulas and identities for bilateral basic hypergeometric series. By employing the bilateral Bailey transformation given by Andrews and Warnaar [1] together with a collection of established summation formulae, we derive several identities of the type rΨr and examine particular limiting cases. The identities obtained here both subsume classical results in the theory of bilateral q-series and constitute a meaningful extension of the Bailey transform machinery to the bilateral setting.
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References
[1] G. E. Andrews and S. O. Warnaar, “The Bailey transform and false theta functions,” Ramanujan J., vol. 14, no. 1, pp. 173–188 (2007), doi: 10.1007/s11139-006-9006-4.
[2] R. Y. Denis, S. N. Singh, and S. P. Singh, “On certain summations and transformations involving basic hypergeometric functions,” Bull. Calcutta Math. Soc., vol. 98, no. 6, pp. 567–578 (2007).
[3] R. Y. Denis, S. N. Singh, and S. P. Singh, “On extended Bailey’s transform,” South East Asian J. Math. Math. Sci., vol. 8, no. 1, pp. 59–66 (2009).
[4] R. Y. Denis, S. N. Singh, and S. P. Singh, “On bilateral Bailey transform,” South East Asian J. Math. Math. Sci., vol. 9, no. 2, pp. 59–67 (2011).
[5] G. Gasper and M. Rahman, Basic Hypergeometric Series, 2nd ed. Cambridge, U.K.: Cambridge University Press (2004).
[6] C. M. Joshi and Y. Vyas, “Extensions of Bailey’s transform and applications,” Int. J. Math. Math. Sci., vol. 2005, pp. 1909–1923 (2005), doi: 10.1155/IJMMS.2005.1909.
[7] S. N. Singh, S. P. Singh, and V. Yadav, “On Bailey’s transform and expansion of basic hypergeometric functions—II,” South East Asian J. Math. Math. Sci., vol. 11, no. 2, pp. 37–46 (2015).
[8] S. P. Singh, S. N. Singh, and V. Yadav, “A note on symmetric bilateral Bailey transform,” South East Asian J. Math. Math. Sci., vol. 15, no. 1, pp. 89–96 (2019).
[9] S. P. Singh and V. Yadav, “On certain transformation formulas involving basic hypergeometric series,” Proc. Natl. Acad. Sci. India Sect. A Phys. Sci., vol. 90, no. 4, pp. 623–627 (2020), doi: 10.1007/s40010-019-00609-4.
[10] H. M. Srivastava, S. N. Singh, S. P. Singh, and V. Yadav, “Certain derived WP-Bailey pairs and transformation formulas for q-hypergeometric series,” Filomat, vol. 31, no. 14, pp. 4619–4628 (2017), doi: 10.2298/FIL1714619S.
[11] A. Verma and V. K. Jain, “Some summation formulae of basic hypergeometric series,” Indian J. Pure Appl. Math., vol. 11, no. 8, pp. 1021–1038 (1980).
[12] A. Verma and V. K. Jain, “Certain transformations of basic hypergeometric series and their applications,” Pacific J. Math., vol. 101, no. 2, pp. 333–349 (1982).
[13] J. Yadav, V. Yadav, and M. Mishra, “On some application of Bailey’s transform in generalized basic hypergeometric transformations,” South East Asian J. Math. Math. Sci., vol. 17, no. 2, pp. 75–82 (2021).
[14] B. V. Zala and J. C. Prajapati, “Certain properties of the Wright function,” J. Interdiscip. Math., vol. 29, no. 4, pp. 909–925 (2026), doi: 10.47974/JIM-2380.
[15] H. Nagar and S. Mishra, “Application of Marichev-Saigo-Maeda fractional differential and integral operators on composition of two special functions,” J. Interdiscip. Math., vol. 27, no. 8, pp. 1869–1874 (2024).
[16] S. Lather and H. Nagar, “A study of fractional kinetic equations associated with k-incomplete second Appell hypergeometric matrix functions,” J. Interdiscip. Math., vol. 29, no. 8, pp. 2421–2430 (2026), doi: 10.47974/JIM-2627.
