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Krattenthaler Matrix Inversion Formula

Let (a)_i and (b)_i be sequences of complex numbers such that b_j!=b_k for j!=k, 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+b_k))/(product_(j=k+1)^(n)(b_j-b_k))

and

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

where the product over an empty set is 1. Then F and G are matrix inverses (Bhatnagar 1995, pp. 16-17).

This result simplifies to the Gould and Hsu matrix inversion formula when b_k=k, to Carlitz's q-analog for b_k=q^k (Carlitz 1972), and specialized to Bressoud's matrix theorem (Bressoud 1983) for b_k=q^(-k)+aq^k and a_k=-(aq^(-j)/b)-bq^j (Bhatnagar 1995, p. 17).

The formula can also be extended to a summation theorem which generalizes Gosper's bibasic sum (Gasper and Rahman 1990, p. 240; Bhatnagar 1995, p. 19).

See also

Bailey's Transformation, Gould and Hsu 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.Bressoud, D. M. "A Matrix Inverse." Proc. Amer. Math. Soc. 88, 446-448, 1983.Carlitz, L. "Some Inverse Relations." Duke Math. J. 40, 893-901, 1972.Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990.Krattenthaler, C. "Operator Methods and Lagrange Inversions: A Unified Approach to Lagrange Formulas." Trans. Amer. Math. Soc. 305, 431-465, 1988.Riordan, J. Combinatorial Identities. New York: Wiley, 1979.

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Krattenthaler Matrix Inversion Formula

Cite this as:

Weisstein, Eric W. "Krattenthaler Matrix Inversion Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KrattenthalerMatrixInversionFormula.html

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