An extension of Warnaar’s matrix inversion

Ma, X.

doi:10.1090/S0002-9939-05-07912-8

An extension of Warnaar’s matrix inversion
HTML articles powered by AMS MathViewer

by X. R. Ma PDF

Proc. Amer. Math. Soc. 133 (2005), 3179-3189 Request permission

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

George E. Andrews, $q$-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra, CBMS Regional Conference Series in Mathematics, vol. 66, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 858826, DOI 10.1090/cbms/066
D. M. Bressoud, A matrix inverse, Proc. Amer. Math. Soc. 88 (1983), no. 3, 446–448. MR 699411, DOI 10.1090/S0002-9939-1983-0699411-9
L. Carlitz, Some inverse relations, Duke Math. J. 40 (1973), 893–901. MR 337651, DOI 10.1215/S0012-7094-73-04083-0
W. C. Chu and L. C. Hsu, Some new applications of Gould-Hsu inversion, J. Combin. Inform. System Sci. 14 (1989), no. 1, 1–4. MR 1068649
Igor B. Frenkel and Vladimir G. Turaev, Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions, The Arnold-Gelfand mathematical seminars, Birkhäuser Boston, Boston, MA, 1997, pp. 171–204. MR 1429892
Ira Gessel and Dennis Stanton, Applications of $q$-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983), no. 1, 173–201. MR 690047, DOI 10.1090/S0002-9947-1983-0690047-7
H. W. Gould and L. C. Hsu, Some new inverse series relations, Duke Math. J. 40 (1973), 885–891. MR 337652, DOI 10.1215/S0012-7094-73-04082-9
Ch. Krattenthaler, Operator methods and Lagrange inversion: a unified approach to Lagrange formulas, Trans. Amer. Math. Soc. 305 (1988), no. 2, 431–465. MR 924765, DOI 10.1090/S0002-9947-1988-0924765-4
C. Krattenthaler, A new matrix inverse, Proc. Amer. Math. Soc. 124 (1996), no. 1, 47–59. MR 1291781, DOI 10.1090/S0002-9939-96-03042-0
Christian Krattenthaler and Michael Schlosser, A new multidimensional matrix inverse with applications to multiple $q$-series, Discrete Math. 204 (1999), no. 1-3, 249–279. MR 1691873, DOI 10.1016/S0012-365X(98)00374-4
Xin Rong Ma, A short proof of Krattenthaler formulas, Acta Math. Sin. (Engl. Ser.) 18 (2002), no. 2, 289–292. MR 1910964, DOI 10.1007/s101140200161
Stephen C. Milne and Gaurav Bhatnagar, A characterization of inverse relations, Discrete Math. 193 (1998), no. 1-3, 235–245. Selected papers in honor of Adriano Garsia (Taormina, 1994). MR 1661371, DOI 10.1016/S0012-365X(98)00143-5
John Riordan, Combinatorial identities, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0231725
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.
Michael Schlosser, Multidimensional matrix inversions and $A_r$ and $D_r$ basic hypergeometric series, Ramanujan J. 1 (1997), no. 3, 243–274. MR 1606918, DOI 10.1023/A:1009705129155
Michael Schlosser, Some new applications of matrix inversions in $A_r$, Ramanujan J. 3 (1999), no. 4, 405–461. MR 1738906, DOI 10.1023/A:1009809424076
V. P. Spiridonov, Theta hypergeometric series, Asymptotic combinatorics with application to mathematical physics (St. Petersburg, 2001) NATO Sci. Ser. II Math. Phys. Chem., vol. 77, Kluwer Acad. Publ., Dordrecht, 2002, pp. 307–327. MR 2000728
S. O. Warnaar, Summation and transformation formulas for elliptic hypergeometric series, Constr. Approx. 18 (2002), no. 4, 479–502. MR 1920282, DOI 10.1007/s00365-002-0501-6
E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759