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An extension of Warnaar's matrix inversion
Author(s):
X.
R.
Ma
Journal:
Proc. Amer. Math. Soc.
133
(2005),
3179-3189.
MSC (2000):
Primary 05A10, 05A19, 33D15;
Secondary 05A15, 33C20, 33D99
Posted:
May 9, 2005
MathSciNet review:
2160179
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Abstract:
We present a necessary and sufficient condition for two matrices given by two bivariate functions to be inverse to each other with certainty in the cases of Krattenthaler formula and Warnaar's elliptic matrix inversion. Immediate consequences of our result are some known functions and a constructive approach to derive new matrix inversions from known ones.
References:
-
- 1.
- G.E.Andrews, q-series: their development and application in analysis,number theory, combinatorics, physics and computer algebra, NSF CBMS Regional Conf. Series, Vol 66, 1986. MR 0858826 (88b:11063)
- 2.
- D.M.Bressoud, A matrix inverse, Proc. Amer. Math. Soc. 88 (1983), 446-448. MR 0699411 (84g:33003)
- 3.
- L.Carlitz, Some inverse relations, Duke Math. J. 40 (1973), 893-901. MR 0337651 (49:2420)
- 4.
- W.C.Chu, L.C.Hsu, Some new applications of Gould-Hsu inversion, J. Combin. Inform. System Sci. 14 (1989), 1-4. MR 1068649 (92c:05018)
- 5.
- I.B.Frenkel, V.G.Turaev, Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions, in V. I. Arnold et al. (eds.), The Arnold-Gelfand Mathematical Seminars, 171-204, Birkhauser, Boston, 1997. MR 1429892 (98k:33034)
- 6.
- I.Gessel, D.Stanton, Applications of q-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983), 173-201. MR 0690047 (84f:33009)
- 7.
- H.W.Gould, L.C.Hsu, Some new inverse series relation, Duke Math. J. 40 (1973), 885-891. MR 0337652 (49:2421)
- 8.
- C.Krattenthaler, Operator methods and Lagrange inversion, a unified approach to Lagrange formulas, Trans. Amer. Math. Soc. 305 (1988), 431-465. MR 0924765 (89d:05017)
- 9.
- C.Krattenthaler, A new matrix inverse, Proc. Amer. Math. Soc. 124 (1996), 47-59. MR 1291781 (96d:15004)
- 10.
- C.Krattenthaler, M.Schlosser, A new multidimensional matrix inversions with applications to multiple
 -series, Discrete Math. 204 (1999), 249-279. MR 1691873 (2000j:33011) - 11.
- Xinrong Ma, A short proof of Krattenthaler formulas, Acta Math. Sin.(Engl. Ser.) 18 (2002), no. 2, 289-292. MR 1910964 (2003b:05009)
- 12.
- S.C.Milne, G.Bhatnagar, A characterization of inverse relations, Discrete Math. 193 (1998), 235-245. MR 1661371 (2000a:05019)
- 13.
- J. Riordan, Combinatorial identities, John Wiley & Sons, Inc., New York-London-Sydney 1968. MR 0231725 (38:53)
- 14.
- H.Rosengren, M.Schlosser, On Warnnar's elliptic matrix inversion and Karlsson-Minton-type elliptic hypergeometric series, J. Comput. Appl. Math. 178 (2005), 377-391.
- 15.
- M.Schlosser, Multidimensional matrix inversions and
 and  basic hypergemetric series, Ramanujan J. 1 (1997), 243-274. MR 1606918 (99b:33036) - 16.
- M.Schlosser, Some new applications of matrix inversions in
 , Ramanujan J. 3 (1999), 405-461. MR 1738906 (2001c:33032) - 17.
- V.P.Spiridonov, Theta hypergeometric series, in V. A. Malyshev and A. M. Vershik (eds.), Asymptotic Combinatorics with Applications to Mathematical Physics, 307-327, Kluwer Acad. Publ., Dordrecht, 2002. MR 2000728 (2004j:33012)
- 18.
- S.O.Warnaar, Summmation and transformation formulas for elliptic hypergeometric series, Constr. Approx. 18 (2002), 479-502. MR 1920282 (2003h:33018)
- 19.
- E.T.Whittaker, G.N.Watson, A course of modern analysis, Reprint of the fourth (1927) edition, Cambridge University Press, Cambridge, 1996. MR 1424469 (97k:01072)
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Additional Information:
X.
R.
Ma
Affiliation:
Department of Mathematics, SuZhou University, SuZhou 215006, People's Republic of China
Email:
xrma@public1.sz.js.cn
DOI:
10.1090/S0002-9939-05-07912-8
PII:
S 0002-9939(05)07912-8
Keywords:
Matrix inversion,
elliptic hypergeometric series,
Lagrange operator,
kernel,
antisymmetric,
addition formula,
polynomial transformation
Received by editor(s):
May 4, 2004
Received by editor(s) in revised form:
May 31, 2004, June 16, 2004 and June 22, 2004
Posted:
May 9, 2005
Communicated by:
John R. Stembridge
Copyright of article:
Copyright
2005,
American Mathematical Society
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