# A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS

Harsh, Harsh Vardhan;Kim, Yong Sup;Rakha, Medhat Ahmed;Rathie, Arjun Kumar

• Published : 2016.01.31
• 13 2

#### Abstract

In 1812, Gauss obtained fifteen contiguous functions relations. Later on, 1847, Henie gave their q-analogue. Recently, good progress has been done in finding more contiguous functions relations by employing results due to Gauss. In 1999, Cho et al. have obtained 24 new and interesting contiguous functions relations with the help of Gauss's 15 contiguous relations. In fact, such type of 72 relations exists and therefore the rest 48 contiguous functions relations have very recently been obtained by Rakha et al.. Thus, the paper is in continuation of the paper [16] published in Computer & Mathematics with Applications 61 (2011), 620.629. In this paper, first we obtained 15 q-contiguous functions relations due to Henie by following a different method and then with the help of these 15 q-contiguous functions relations, we obtain 72 new and interesting q-contiguous functions relations. These q-contiguous functions relations have wide applications.

#### Keywords

basic hypergeometric series;q-contiguous functions relations;Gauss's contiguous functions relations

#### References

1. G. E. Andrews, On the q-analog of Kummer's theorem and applications, Duke Math. J. 40 (1973), 525-528. https://doi.org/10.1215/S0012-7094-73-04045-3
2. W. N. Bailey, A note on certain q-identities, Quart. J. Math. 12 (1941), 173 - 175.
3. Y. J. Cho, T. Y. Seo, and J. Choi, Note on contiguous functions relations, East Asian Math. J. 15 (1999), no. 1, 29-38.
4. J. A. Dhaum, The basic analog of Kummer's theorems, Bull. Amer. Math. Soc. 8 (1942), 711-713.
5. G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge University Press, 1990.
6. C. F. Gauss, Disquisitiones generales circa seriem infinitam . . . , Comm. Soc. Reg. Sci. Gott. Rec., Vol. II; reprinted in Werke 3 (1876), 123-162.
7. E. Henie, Untersuchungen uber die Rehie, J. Reine Angew. Math. 34 (1847), 285-328.
8. A. K. Ibrahim and M. A. Rakha, Contiguous relations and their computations for $_2F_1$ hypergeometric series, Comput. Math. Appl. 56 (2008), no. 8, 1918-1926. https://doi.org/10.1016/j.camwa.2008.04.018
9. F. H. Jackson, Transformation of q-series, Messenger of Math. 39 (1910), 145-153.
10. Y. S. Kim and A. K. Rathie, Another method for proving a q-analogue of Gauss's summation theorem, Far East J. Math. Sci. 5 (2002), no. 3, 317-322.
11. Y. S. Kim, A. K. Rathie, and J. Choi, Three term contiguous functional relations for basic hypergeometric series $_2{\Phi}_1$, Commun. Korean Math. Soc. 20 (2005), no. 2, 395-403. https://doi.org/10.4134/CKMS.2005.20.2.395
12. Y. S. Kim, A. K. Rathie, and C. H. Lee, On q-analog of Kummer's theorem and its contiguous results, Commun. Korean Math. Soc. 18 (2003), no. 1, 151-157. https://doi.org/10.4134/CKMS.2003.18.1.151
13. W. Miller, Jr., Lie theory and generalizations of hypergeometric functions, SIAM J. Appl. Math. 25 (1973), 226-235. https://doi.org/10.1137/0125026
14. E. D. Rainville, The contiguous function relations for $_pF_q$ with applications to Bateman's $J_n^{u,v}$ and Rice's $H_n({\zeta},p,v)$, Bull. Amer. Math. Soc. 51 (1945), 714-723. https://doi.org/10.1090/S0002-9904-1945-08425-0
15. E. D. Rainville, Special Functions, The Macmillan Company, New York, 1960.
16. M. A. Rakha, A. K. Rathie, and P. Chopra, On contiguous function relations, Comput. Math. Appl. 61 (2011), 620-629. https://doi.org/10.1016/j.camwa.2010.12.008
17. C. Wei and D. Gong, q-Extensions of Gauss' fifteen contiguous relation for $_2F_1$ series, Commun. Computer Information Science 105 (2011), no. 2, 85-92.