## A new matrix inverse.(English)Zbl 0843.15005

The inverse of a specific infinite-dimensional matrix is computed, thereby unifying a number of previous matrix inversions. The inversion theorem is applied to derive a number of summation formulas of hypergeometric type.
Reviewer: G.Bonanno (Davis)

### MSC:

 15A09 Theory of matrix inversion and generalized inverses 40C05 Matrix methods for summability
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### References:

 [1] 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. [2] W. N. Bailey, Some identities in combinatory analysis, Proc. London Math. Soc. (2) 49 (1947), 421–435. · Zbl 0041.03403 [3] ——, Identities of the Roger–Ramanujan type, Proc. London Math. Soc. (2) 50 (1949), 1–10. [4] D. M. Bressoud, Some identities for terminating \?-series, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 2, 211 – 223. · Zbl 0454.33003 [5] D. M. Bressoud, A matrix inverse, Proc. Amer. Math. Soc. 88 (1983), no. 3, 446 – 448. · Zbl 0529.05006 [6] L. Carlitz, Some inverse relations, Duke Math. J. 40 (1973), 893 – 901. · Zbl 0276.05012 [7] Интеграл$$^{\приме}$$ное представление и вычисление комбинаторных сумм., Издат. ”Наука” Сибирск. Отдел., Новосибирск, 1977 (Руссиан). · Zbl 0453.05001 [8] George Gasper, Summation, transformation, and expansion formulas for bibasic series, Trans. Amer. Math. Soc. 312 (1989), no. 1, 257 – 277. · Zbl 0664.33010 [9] 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. · Zbl 0707.33009 [10] 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. · Zbl 0695.33001 [11] Ira Gessel and Dennis Stanton, Applications of \?-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983), no. 1, 173 – 201. · Zbl 0513.33001 [12] H. W. Gould, A series transformation for finding convolution identities, Duke Math. J. 28 (1961), 193 – 202. · Zbl 0122.30501 [13] H. W. Gould, A new convolution formula and some new orthogonal relations for inversion of series, Duke Math. J. 29 (1962), 393 – 404. · Zbl 0122.30502 [14] H. W. Gould, A new series transform with applications to Bessel, Legendre, and Tchebycheff polynomials, Duke Math. J. 31 (1964), 325 – 334. · Zbl 0137.04003 [15] H. W. Gould, Inverse series relations and other expansions involving Humbert polynomials, Duke Math. J. 32 (1965), 697 – 711. · Zbl 0135.12001 [16] H. W. Gould and L. C. Hsu, Some new inverse series relations, Duke Math. J. 40 (1973), 885 – 891. · Zbl 0281.05008 [17] 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. · Zbl 0668.00003 [18] Ch. Krattenthaler, Operator methods and Lagrange inversion: a unified approach to Lagrange formulas, Trans. Amer. Math. Soc. 305 (1988), no. 2, 431 – 465. · Zbl 0653.05007 [19] Mizan Rahman, Some quadratic and cubic summation formulas for basic hypergeometric series, Canad. J. Math. 45 (1993), no. 2, 394 – 411. · Zbl 0774.33012 [20] M. Rahman, Some cubic summation formulas for basic hypergeometric series, Utilitas Math. 36 (1989), 161 – 172. · Zbl 0691.33002 [21] John Riordan, Combinatorial identities, John Wiley & Sons, Inc., New York-London-Sydney, 1968. · Zbl 0194.00502 [22] D. Singer, $$q$$-analogues of Lagrange inversion, Ph.D. Thesis, Univ. of California, San Diego, CA, 1992. · Zbl 0842.33009 [23] Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. [24] 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. · Zbl 0488.33001
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