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Gould and Hsu Matrix Inversion Formula

Let (a)_i be a sequence of complex numbers and let the lower triangular matrices F=(f)_(nk) and G=(g)_(nk) be defined as

f_(nk)=(product_(j=k)^(n-1)(a_j+k))/((n-k)!)

and

g_(nk)=(-1)^(n-k)(a_k+k)/(a_n+n)(product_(j=k+1)^(n)(a_j+n))/((n-k)!),

where the product over an empty set is 1. Then F and G are matrix inverses (Bhatnagar 1995, pp. 15-16 and 50-51). The Krattenthaler matrix inversion formula is a generalization of this result.

See also

Bailey's Transformation, Krattenthaler Matrix Inversion Formula

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References

Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University, 1995. http://www.math.ohio-state.edu/~milne/papers/Gaurav.whole.thesis.7.4.ps.Carlitz, L. "Some Inverse Relations." Duke Math. J. 40, 893-901, 1972.Chu, W. C. and Hsu, L. C. "Some New Applications of Gould-Hsu Inversions." J. Combin. Inform. System Sci. 14, 1-4, 1989.Gessel, I. and Stanton, D. "Application of q-Lagrange Inversion to Basic Hypergeometric Series." Trans. Amer. Math. Soc. 277, 173-201, 1983.Gould, H. W. and Hsu, L. C. "Some New Inverse Series Relations." Duke Math. J. 40, 885-891, 1973.Riordan, J. Combinatorial Identities. New York: Wiley, 1979.

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Gould and Hsu Matrix Inversion Formula

Cite this as:

Weisstein, Eric W. "Gould and Hsu Matrix Inversion Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GouldandHsuMatrixInversionFormula.html

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