A new matrix inverse

Krattenthaler, C.

doi:10.1090/S0002-9939-96-03042-0

A new matrix inverse
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Proc. Amer. Math. Soc. 124 (1996), 47-59 Request permission

Abstract:

We compute the inverse of a specific infinite-dimensional matrix, thus unifying a number of previous matrix inversions. Our inversion theorem is applied to derive a number of summation formulas of hypergeometric type.

References

George E. Andrews, Connection coefficient problems and partitions, Relations between combinatorics and other parts of mathematics (Proc. Sympos. Pure Math., Ohio State Univ., Columbus, Ohio, 1978) Proc. Sympos. Pure Math., XXXIV, Amer. Math. Soc., Providence, R.I., 1979, pp. 1–24. MR 525316
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
D. M. Bressoud, Some identities for terminating $q$-series, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 2, 211–223. MR 600238, DOI 10.1017/S0305004100058114
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
G. P. Egorychev, Integral′noe predstavlenie i vychislenie kombinatornykh summ, Izdat. “Nauka” Sibirsk. Otdel., Novosibirsk, 1977 (Russian). MR 0491209
George Gasper, Summation, transformation, and expansion formulas for bibasic series, Trans. Amer. Math. Soc. 312 (1989), no. 1, 257–277. MR 953537, DOI 10.1090/S0002-9947-1989-0953537-0
George Gasper and Mizan Rahman, An indefinite bibasic summation formula and some quadratic, cubic and quartic summation and transformation formulas, Canad. J. Math. 42 (1990), no. 1, 1–27. MR 1043508, DOI 10.4153/CJM-1990-001-5
George Gasper and Mizan Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR 1052153
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, A series transformation for finding convolution identities, Duke Math. J. 28 (1961), 193–202. MR 123895
H. W. Gould, A new convolution formula and some new orthogonal relations for inversion of series, Duke Math. J. 29 (1962), 393–404. MR 139906, DOI 10.1215/S0012-7094-62-02938-1
H. W. Gould, A new series transform with applications to Bessel, Legendre, and Tchebycheff polynomials, Duke Math. J. 31 (1964), 325–334. MR 161063, DOI 10.1215/S0012-7094-64-03131-X
H. W. Gould, Inverse series relations and other expansions involving Humbert polynomials, Duke Math. J. 32 (1965), 697–711. MR 188510, DOI 10.1215/S0012-7094-65-03275-8
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
Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1989. A foundation for computer science. MR 1001562
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
Mizan Rahman, Some quadratic and cubic summation formulas for basic hypergeometric series, Canad. J. Math. 45 (1993), no. 2, 394–411. MR 1208123, DOI 10.4153/CJM-1993-020-8
M. Rahman, Some cubic summation formulas for basic hypergeometric series, Utilitas Math. 36 (1989), 161–172. MR 1030781
John Riordan, Combinatorial identities, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0231725
D. Singer, $q$-analogues of Lagrange inversion, Ph.D. Thesis, Univ. of California, San Diego, CA, 1992.
Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688
A. Verma and V. K. Jain, Transformations of nonterminating basic hypergeometric series, their contour integrals and applications to Rogers-Ramanujan identities, J. Math. Anal. Appl. 87 (1982), no. 1, 9–44. MR 653604, DOI 10.1016/0022-247X(82)90151-2