Summation, transformation, and expansion formulas for bibasic series
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- by George Gasper PDF
- Trans. Amer. Math. Soc. 312 (1989), 257-277 Request permission
Abstract:
An indefinite bibasic sum containing three parameters is evaluated and used to derive bibasic extensions of Eulerβs transformation formula and of a Fields and Wimp expansion formula. It is also used to derive a transformation formula involving four independent bases, a $q$-Lagrange inversion formula, and some quadratic, cubic and quartic summation formulas.References
- R. P. Agarwal and Arun Verma, Generalized basic hypergeometric series with unconnected bases, Proc. Cambridge Philos. Soc. 63 (1967), 727β734. MR 212216, DOI 10.1017/s0305004100041724
- R. P. Agarwal and Arun Verma, Generalized basic hypergeometric series with unconnected bases. II, Quart. J. Math. Oxford Ser. (2) 18 (1967), 181β192. MR 214815, DOI 10.1093/qmath/18.1.181
- W. A. Al-Salam and A. Verma, On quadratic transformations of basic series, SIAM J. Math. Anal. 15 (1984), no.Β 2, 414β421. MR 731877, DOI 10.1137/0515032
- G. E. Andrews, On basic hypergeometric series, mock theta functions, and partitions. I, Quart. J. Math. Oxford Ser. (2) 17 (1966), 64β80. MR 193282, DOI 10.1093/qmath/17.1.64
- Richard Askey and George Gasper, Inequalities for polynomials, The Bieberbach conjecture (West Lafayette, Ind., 1985) Math. Surveys Monogr., vol. 21, Amer. Math. Soc., Providence, RI, 1986, pp.Β 7β32. MR 875228, DOI 10.1090/surv/021/02 W. N. Bailey, Generalized hypergeometric series, Cambridge Univ. Press, Cambridge, 1935.
- Louis de Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), no.Β 1-2, 137β152. MR 772434, DOI 10.1007/BF02392821
- D. M. Bressoud, The Bailey lattice: an introduction, Ramanujan revisited (Urbana-Champaign, Ill., 1987) Academic Press, Boston, MA, 1988, pp.Β 57β67. MR 938960
- L. Carlitz, Some inverse relations, Duke Math. J. 40 (1973), 893β901. MR 337651
- Jerry L. Fields and Mourad E. H. Ismail, Polynomial expansions, Math. Comp. 29 (1975), 894β902. MR 372472, DOI 10.1090/S0025-5718-1975-0372472-6
- Jerry L. Fields and Jet Wimp, Expansions of hypergeometric functions in hypergeometric functions, Math. Comp. 15 (1961), 390β395. MR 125992, DOI 10.1090/S0025-5718-1961-0125992-3
- George Gasper, A short proof of an inequality used by de Branges in his proof of the Bieberbach, Robertson and Milin conjectures, Complex Variables Theory Appl. 7 (1986), no.Β 1-3, 45β50. MR 877650, DOI 10.1080/17476938608814185
- George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR 2128719, DOI 10.1017/CBO9780511526251
- Ira Gessel and Dennis Stanton, Strange evaluations of hypergeometric series, SIAM J. Math. Anal. 13 (1982), no.Β 2, 295β308. MR 647127, DOI 10.1137/0513021
- Ira Gessel and Dennis Stanton, Applications of $q$-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983), no.Β 1, 173β201. MR 690047, DOI 10.1090/S0002-9947-1983-0690047-7
- Ira Gessel and Dennis Stanton, Another family of $q$-Lagrange inversion formulas, Rocky Mountain J. Math. 16 (1986), no.Β 2, 373β384. MR 843058, DOI 10.1216/RMJ-1986-16-2-373 F. H. Jackson, Transformations of $q$-series, Messenger of Math. 39 (1910), 145-153. Y. L. Luke, The special functions and their applications. II, Academic Press, New York, 1969.
- Mizan Rahman, Some quadratic and cubic summation formulas for basic hypergeometric series, Canad. J. Math. 45 (1993), no.Β 2, 394β411. MR 1208123, DOI 10.4153/CJM-1993-020-8
- D. B. Sears, On the transformation theory of basic hypergeometric functions, Proc. London Math. Soc. (2) 53 (1951), 158β180. MR 41981, DOI 10.1112/plms/s2-53.2.158
- Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688
- H. M. Srivastava, Certain $q$-polynomial expansions for functions of several variables, IMA J. Appl. Math. 30 (1983), no.Β 3, 315β323. MR 719983, DOI 10.1093/imamat/30.3.315
- H. M. Srivastava, Certain $q$-polynomial expansions for functions of several variables. II, IMA J. Appl. Math. 33 (1984), no.Β 2, 205β209. MR 767521, DOI 10.1093/imamat/33.2.205
- Dennis Stanton, Recent results for the $q$-Lagrange inversion formula, Ramanujan revisited (Urbana-Champaign, Ill., 1987) Academic Press, Boston, MA, 1988, pp.Β 525β536. MR 938977
- Arun Verma, Some transformations of series with arbitrary terms, Ist. Lombardo Accad. Sci. Lett. Rend. A 106 (1972), 342β353. MR 328144
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 312 (1989), 257-277
- MSC: Primary 33A70; Secondary 33A35
- DOI: https://doi.org/10.1090/S0002-9947-1989-0953537-0
- MathSciNet review: 953537