q-Series Identities and Reducibility of Basic Double Hypergeometric Functions

Srivastava, H. M. ; Jain, V. K.

Canadian journal of mathematics, Tome 38 (1986) no. 1, pp. 215-231

Résumé

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.

Détail

DOI : 10.4153/CJM-1986-010-3

Bibliographie

[1] 1. Agarwal, R. P., Some basic hypergeometric identities, Ann. Soc. Sci. Bruxelles Sér. I 67 (1953), 186–202. Google Scholar

[2] 2. Agarwal, R. P., Some relations between basic hypergeometric functions of two variables, Rend. Circ. Mat. Palermo 3 (1954), 76–82. Google Scholar

[3] 3. Andrews, G. E., Summations and transformations for basic Appell series, J. London Math. Soc. (2) 4 (1972), 618–622. Google Scholar

[4] 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] 5. Agarwal, R. P., On q-analogues of the Watson and Whipple summations, SIAM J. Math. Anal. 7 (1976), 332–336. Google Scholar

[6] 6. Appell, P. et de Fériet, J. Kampé, Fonctions hypergéométriques et hypersphériques; polynômes d'Hermite (Gauthier-Villars, Paris, 1926). Google Scholar

[7] 7. Bailey, W. N., Products of generalized hypergeometric series, Proc. London Math. Soc. (2) 25 (1928), 242–254. Google Scholar

[8] 8. Bailey, W. N., On the sum of a terminating F(1), Quart. J. Math. Oxford Ser. (2) 4 (1953), 237–240. Google Scholar

[9] 9. Burchnall, J. L., Differential equations associated with hypergeometric functions, Quart. J. Math. Oxford Ser. 13 (1942), 90–106. Google Scholar

[10] 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] 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] 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] 13. Carlitz, L., Some formulas of F. H. Jackson, Monatsh. Math. 73 (1969), 193–198. Google Scholar

[14] 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] 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] 16. Exton, H., Multiple hypergeometric functions and applications (Halsted Press, Ellis Horwood, Chichester), (John Wiley &amp; Sons, New York, London, Sydney and Toronto, 1976). Google Scholar

[17] 17. Exton, H., The basic double hypergeometric transforms, Indian J. Math. 19 (1977), 35–40. Google Scholar

[18] 18. Exton, H., q-hypergeometric functions and applications (Halsted Press, Ellis Horwood, Chichester), (John Wiley &amp; Sons, New York, Brisbane, Chichester and Toronto, 1983). Google Scholar

[19] 19. Jackson, F. H., Transformations of q-series, Messenger Math. 39 (1910), 145–153. Google Scholar

[20] 20. Jackson, F. H., On basic double hypergeometric functions, Quart. J. Math. Oxford Ser. 13 (1942), 69–82. Google Scholar

[21] 21. Jackson, F. H., Basic double hypergeometric functions (II), Quart. J. Math. Oxford Ser. 15 (1944), 49–61. Google Scholar

[22] 22. Jain, V. K., Some expansions involving basic hypergeometric functions of two variables, Pacific J. Math. 91 (1980), 349–361. Google Scholar

[23] 23. Jain, V. K., Some transformations of basic hyper geometric functions. Part II, SIAM J. Math. Anal. 12 (1981), 957–961. Google Scholar

[24] 24. Rainville, E. D., Special functions (Macmillan, New York, 1960). Reprinted by Chelsea Publ. Co., Bronx, New York (1971). Google Scholar

[25] 25. Slater, L. J., Some basic hyper geometric transforms, J. London Math. Soc. 30 (1955), 351–360. Google Scholar

[26] 26. Slater, L. J., Generalized hyper geometric functions (Cambridge Univ. Press, Cambridge, London and New York, 1966). Google Scholar

[27] 27. Srivastava, H. M., On the reducibility of’ Appell's function F, Canad. Math. Bull. 16 (1973), 295–298. Google Scholar

[28] 28. Srivastava, H. M., Some generalizations of Carlson's identity, Boll. Un. Mat. Ital. 18A (1981), 138–143. Google Scholar

[29] 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] 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] 31. Upadhyay, M., q-fractional differentiation and basic hyper geometric transformations, Ann. Polon. Math. 25 (1971), 109–124. Google Scholar

[32] 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] 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] 34. Verma, A., A quadratic transformation of a basic hyper geometric series, SIAM J. Math. Anal. 11 (1980), 425–427. Google Scholar

[35] 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] 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