Abstract
We have studied the asymptotics of two special two-matrix hypergeometric functions. The validity of the asymptotic expressions for these functions is seen in several selected numerical comparisons between the exact and asymptotic results. These hypergeometric functions find applications in configuration statistics of macromolecules as well as multivariate statistics.
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References
Bingham, C. (1974). An identity involving partitional generalized binomial coefficients,J. Multivariate Anal.,4, 210–223.
Coriell, S. R. and Jackson, J. L. (1967). Probability distribution of the radius of gyration of a flexible polymer,J. Math. Phys.,8, 1276–1284.
deGennes, P. G. (1979).Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, New York.
Doi, M. and Edwards, S. F. (1986).The Theory of Polymer Dynamics, Clarendon Press, Oxford.
Eichinger, B. E. (1972). Elasticity theory. I: distribution functions for perfect phantom networks,Macromolecules,5, 496–505.
Eichinger, B. E. (1977).An approach to distribution functions for Gaussian molecules, Macromolecules,10, 671–675.
Eichinger, B. E. (1980). Configuration statistics of Gaussian molecules,Macromolecules,13, 1–11.
Eichinger, B. E. (1983). The theory of high elasticity,Annual Reviews of Physical Chemistry,34, 359–387.
Eichinger, B. E. (1985). Shape distributions for Gaussian molecules,Macromolecules,18, 211–216.
Eichinger, B. E., Shy, L. Y. and Wei, G. (1989). Distribution functions in elasticity,Die Makromolecular chemie, Macromolecular Symposia,30, 237–249.
Fixman, M. (1962). Radius of gyration of polymer chains,Journal of Chemical Physics,36, 306–310.
Flory, P. J. (1953).Principles of Polymer Chemistry, Cornell University Press, Ithaca, New York.
Flory, P. J. (1969).Statistical Mechanics of Chain Molecules, Wiley, New York.
Fujita, H. and Norisuye, T. (1970). Some topics concerning the radius of gyration of linear polymer molecules in solution,Journal of Chemical Physics,52, 1115–1120.
Hsu, L. C. (1948). A theorem on the asymptotic behavior of a multiple integral,Duke Math. J.,15, 623–632.
James, A. T. (1964). Distributions of matrix variables and latent roots derived from normal samples,Ann. Math. Statist.,35, 475–501.
James, A. T. (1969). Test of equality of the latent roots of the covariance matrix,Multivariate Analysis (ed. P. R. Krishnaiah), 205–218. Academic Press, New York.
Martin, J. E. and Eichinger, B. E. (1978). Distribution functions for Gaussian molecules. I: stars and random regular nets,Journal of Chemical Physics,69, 4588–4594.
Muirhead, R. J. (1978). Latent roots and matrix variates: a review of some asymptotic results,Ann. Statist.,6, 5–33.
Muirhead, R. J. (1982).Aspects of Multivariate Statistical Theory, Wiley, New York.
Shy, L. Y. and Eichinger, B. E. (1986). Shape distributions for Gaussian molecules: circular and linear chains in two dimensions,Macromolecules,19, 838–843.
Stockmayer, W. H. (1974). Statistics of macromolecular shape,XXIVth International Congress of Pure and Applied Chemistry, 91–98, Butterworths, London.
Šolc, K. (1971). Shape of a random-flight chain,Journal of Chemical Physics,55, 335–344.
Šolc, K. (1972). Statistical mechanics of random-flight chains. III: exact square radii distributions for rings,Macromolecules,5, 705–708.
Šolc, K. and Gobush, W. (1974). Statistical mechanics of random-flight chain. VI: distribution of principal components of the radius of gyration for two-dimensional rings,Macromolecules,7, 814–823.
Šolc, K. and Stockmayer, W. H. (1971).Shape of a random-flight chain, Journal of Chemical Physics,54, 2756–2757.
Yamakawa, H. (1971).Modern Theory of Polymer Solution, Harper and Row, New York.
Wei, G. (1989). Distribution function of the radius of gyration for Gaussian molecules,Journal of Chemical Physics,90, 5873–5877.
Wei, G. (1990a). Shape distributions for randomly coiled molecules, Ph.D. dissertation, University of Washington, Seattle.
Wei, G. (1990b). A multidimensional integral,SIAM Rev.,32, p. 479.
Wei, G. and Eichinger, B. E. (1989). Shape distributions for Gaussian molecules: circular and linear chains as spheres and ellipsoids of revolution,Macromolecules,22, 3429–3435.
Wei, G. and Eichinger, B. E. (1990a). Evaluations of distribution functions for flexible macromolecules by the saddle-point method,J. Math. Phys.,31, 1274–1279.
Wei, G. and Eichinger, B. E. (1990b). Shape distributions for Gaussian molecules: circular and linear chains as asymmetric ellipsoids,Macromolecules,23, 4845–4854.
Wei, G. and Eichinger, B. E. (1990c). On shape asymmetry of Gaussian molecules,Journal of Chemical Physics,93, 1430–1435.
Wei, G. and Eichinger, B. E. (1991). Distributions and characterizations of size and shape for randomly coiled molecules,Journal of Computational Polymer Science,1, 41–50.
Zimm, B. H. (1988). Size fluctuations can explain anomalous mobility in field-inverse electrophoresis of DNA,Phys. Rev. Lett.,61, 2965–2968.
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This work was supported by grant DE-FG06-84ER45123 from the Department of Energy, U.S.A.
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Wei, G., Eichinger, B.E. Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules. Ann Inst Stat Math 45, 467–475 (1993). https://doi.org/10.1007/BF00773349
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DOI: https://doi.org/10.1007/BF00773349