Some Certain Summation Formula of q- Series and q–Continued Fractions

Vidhi Bhardwaj; Dr. Jayprakash Yadav

doi:https://doi.org/10.14445/22315373/IJMTT-V66I1P520

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 66 | Issue 1 | Year 2020 | Article Id. IJMTT-V66I1P520 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I1P520

Some Certain Summation Formula of q- Series and q–Continued Fractions

Vidhi Bhardwaj, Dr. Jayprakash Yadav

Citation :

Vidhi Bhardwaj, Dr. Jayprakash Yadav, "Some Certain Summation Formula of q- Series and q–Continued Fractions," International Journal of Mathematics Trends and Technology (IJMTT), vol. 66, no. 1, pp. 165-168, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I1P520

Abstract

In this paper author try to establish transformation formulas using hyper geometric functions. In order to derive these transformations, two well-known methods are used i.e., the q-series and q-continued factions. Main objective is to establish transformation formulas using hyper geometric functions with the help of known transformations formulas in hyper geometric functions. Some more transformation formulas can be established using known hyper geometric functions, continued fractions of two hyper geometric functions can be established with the help of q- fractional operators.The focus of this paper is on hyper geometric functions, which are special functions and solution of a specific second order linear differential equation. We express these hyper geometric functions in terms of their integral representations.Over the years researchers have worked upon some approximations of these interesting integrals, given one of the parameters in the hyper geometric function is large. However there was no unified analysis of all the cases of these integrals.

Keywords

q-series; continued fraction identities.

References

[1] Bailey, W. N. "Some Identities Involving Generalized Hypergeometric Series." Proc. London Math. Soc. 29, 503-516, 1929.
[2] Bailey, W. N. Generalized Hypergeometric Series. Cambridge, England: University Press, 1935.
[3] Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, 1995. http://www.math.ohio-state.edu/~milne/papers/Gaurav.whole.thesis.7.4.ps.
[4] Milne, S. C. and Lilly, G. M. "The A_i and C_i Bailey Transform and Lemma." Bull. Amer. Math. Soc. 26, 258-263, 1992.
[5] An analytic generalization of the Rogers-Ramanujan identities for odd moduli, Proc. Natl. Acad. Sci. USA 71 (1974), 4082–4085.
[6] Problems and prospects for basic hypergeometric functions, Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) (R. Askey, ed.), Math. Res. Center, Univ. Wisconsin, Publ. no. 35, Academic Press, New York, 1975, pp. 191–224.
[7] The theory of partitions, Encyclopedia of Mathematics and Its Applications, vol. 2, Addison-Wesley Publishing, Massachusetts, 1976.
[8] Partitions and Durfee dissection, Amer. J. Math. 101 (1979), no. 3, 735–742.
[9] The hard-hexagon model and Rogers-Ramanujan type identities, Proc. Natl. Acad. Sci. USA 78 (1981), no. 9, part 1, 5290–5292.
[10] Multiple series Rogers-Ramanujan type identities, Pacific J. Math. 114 (1984), no. 2, 267–283.