Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-06-02T16:53:18.424Z Has data issue: false hasContentIssue false
  • English
  • Français

q-Series Identities and Reducibility of Basic Double Hypergeometric Functions

Published online by Cambridge University Press:  20 November 2018

H. M. Srivastava  and
V. K. Jain
H. M. Srivastava
Affiliation:
University of Victoria, Victoria, British Columbia
V. K. Jain
Affiliation:
Bareilly College, Bareilly, Uttar Pradesh, India
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For real or complex q, |q| < 1, let

(1.1)

for arbitrary λ and μ, so that

(1.2)

and

(1.3)

Define, as usual, a generalized q-hypergeometric function by (cf. [26, Chapter 3]; see also [18])

(1.4)

where, for convergence, |q| < 1 and |z| < ∞ when r is a positive integer, or |z| < 1 when r = 0, provided that no zeros appear in the denominator.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 1 , 01 February 1986 , pp. 215 - 231
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Agarwal, R. P., Some basic hypergeometric identities, Ann. Soc. Sci. Bruxelles Sér. I 67 (1953), 186–202.Google Scholar
2. Agarwal, R. P., Some relations between basic hypergeometric functions of two variables, Rend. Circ. Mat. Palermo 3 (1954), 76–82.Google Scholar
3. Andrews, G. E., Summations and transformations for basic Appell series, J. London Math. Soc. (2) 4 (1972), 618–622.Google Scholar
4. Agarwal, R. P., Problems and prospects for basic hypergeometric functions, In Theory and application of special functions (Academic Press, New York, San Franscisco and London, 1975), 191–224.Google Scholar
5. Agarwal, R. P., On q-analogues of the Watson and Whipple summations, SIAM J. Math. Anal. 7 (1976), 332–336.Google Scholar
6. Appell, P. et de Fériet, J. Kampé, Fonctions hypergéométriques et hypersphériques; polynômes d'Hermite (Gauthier-Villars, Paris, 1926).Google Scholar
7. Bailey, W. N., Products of generalized hypergeometric series, Proc. London Math. Soc. (2) 25 (1928), 242–254.Google Scholar
8. Bailey, W. N., On the sum of a terminating 3F2(1), Quart. J. Math. Oxford Ser. (2) 4 (1953), 237–240.Google Scholar
9. Burchnall, J. L., Differential equations associated with hypergeometric functions, Quart. J. Math. Oxford Ser. 13 (1942), 90–106.Google Scholar
10. Burchnall, J. L. and Chaundy, T. W., Expansions of Appell's double hypergeometric functions, Quart. J. Math. Oxford Ser. 11 (1940), 249–270.Google Scholar
11. Burchnall, J. L. and Chaundy, T. W., Expansions of AppelVs double hypergeometric functions (II), Quart. J. Math. Oxford Ser. 12 (1941), 112–128.Google Scholar
12. Buschman, R. G. and Srivastava, H. M., Series identities and reducibility of Kampé de Fériet functions, Math. Proc. Cambridge Philos. Soc. 91 (1982), 435–440.Google Scholar
13. Carlitz, L., Some formulas of F. H. Jackson, Monatsh. Math. 73 (1969), 193–198.Google Scholar
14. Carlson, B. C., The need for a new classification of double hypergeometric series, Proc. Amer. Math. Soc. 56 (1976), 221–224.Google Scholar
15. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher transcendental functions, Vol. I (McGraw-Hill, New York, Toronto and London, 1953).Google Scholar
16. Exton, H., Multiple hypergeometric functions and applications (Halsted Press, Ellis Horwood, Chichester), (John Wiley & Sons, New York, London, Sydney and Toronto, 1976).Google Scholar
17. Exton, H., The basic double hypergeometric transforms, Indian J. Math. 19 (1977), 35–40.Google Scholar
18. Exton, H., q-hypergeometric functions and applications (Halsted Press, Ellis Horwood, Chichester), (John Wiley & Sons, New York, Brisbane, Chichester and Toronto, 1983).Google Scholar
19. Jackson, F. H., Transformations of q-series, Messenger Math. 39 (1910), 145–153.Google Scholar
20. Jackson, F. H., On basic double hypergeometric functions, Quart. J. Math. Oxford Ser. 13 (1942), 69–82.Google Scholar
21. Jackson, F. H., Basic double hypergeometric functions (II), Quart. J. Math. Oxford Ser. 15 (1944), 49–61.Google Scholar
22. Jain, V. K., Some expansions involving basic hypergeometric functions of two variables, Pacific J. Math. 91 (1980), 349–361.Google Scholar
23. Jain, V. K., Some transformations of basic hyper geometric functions. Part II, SIAM J. Math. Anal. 12 (1981), 957–961.Google Scholar
24. Rainville, E. D., Special functions (Macmillan, New York, 1960). Reprinted by Chelsea Publ. Co., Bronx, New York (1971).Google Scholar
25. Slater, L. J., Some basic hyper geometric transforms, J. London Math. Soc. 30 (1955), 351–360.Google Scholar
26. Slater, L. J., Generalized hyper geometric functions (Cambridge Univ. Press, Cambridge, London and New York, 1966).Google Scholar
27. Srivastava, H. M., On the reducibility of’ Appell's function F4, Canad. Math. Bull. 16 (1973), 295–298.Google Scholar
28. Srivastava, H. M., Some generalizations of Carlson's identity, Boll. Un. Mat. Ital. 18A (1981), 138–143.Google Scholar
29. Srivastava, H. M., An elementary proof of Bailey's bilinear generating function for Jacobi polynomials and of its q-analogue, IMA J. Appl. Math. 29 (1982), 275–280.Google Scholar
30. Srivastava, H. M., Certain q-polynomial expansions for functions of several variables. I and II, IMA J. Appl. Math. 30 (1983), 315–323; Ibid. 33 (1984), 205–209.Google Scholar
31. Upadhyay, M., q-fractional differentiation and basic hyper geometric transformations, Ann. Polon. Math. 25 (1971), 109–124.Google Scholar
32. Upadhyay, M., Fractional q-integration and integral representations of basic double hyper geometric series of higher order, Ganita 23 (1972), 19–29.Google Scholar
33. Upadhyay, M., Certain transformations for basic double hyper geometric functions of higher order, Indian J. Pure Appl. Math. 4 (1973), 341–354.Google Scholar
34. Verma, A., A quadratic transformation of a basic hyper geometric series, SIAM J. Math. Anal. 11 (1980), 425–427.Google Scholar
35. Verma, A. and Jain, V. K., Some summation formulae for basic hyper geometric series, Indian J. Pure Appl. Math. 11 (1980), 1021–1038.Google Scholar
36. Verma, A. and Jain, V. K., Transformations between basic hyper geometric series on different bases and identities of Rogers-Ramanujan type, J. Math. Anal. Appl. 76 (1980), 230–269.Google Scholar