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Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules

  • Asymptotic Expansions
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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.

    Google Scholar 

  • 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.

    Google Scholar 

  • deGennes, P. G. (1979).Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, New York.

    Google Scholar 

  • Doi, M. and Edwards, S. F. (1986).The Theory of Polymer Dynamics, Clarendon Press, Oxford.

    Google Scholar 

  • Eichinger, B. E. (1972). Elasticity theory. I: distribution functions for perfect phantom networks,Macromolecules,5, 496–505.

    Google Scholar 

  • Eichinger, B. E. (1977).An approach to distribution functions for Gaussian molecules, Macromolecules,10, 671–675.

    Google Scholar 

  • Eichinger, B. E. (1980). Configuration statistics of Gaussian molecules,Macromolecules,13, 1–11.

    Google Scholar 

  • Eichinger, B. E. (1983). The theory of high elasticity,Annual Reviews of Physical Chemistry,34, 359–387.

    Google Scholar 

  • Eichinger, B. E. (1985). Shape distributions for Gaussian molecules,Macromolecules,18, 211–216.

    Google Scholar 

  • Eichinger, B. E., Shy, L. Y. and Wei, G. (1989). Distribution functions in elasticity,Die Makromolecular chemie, Macromolecular Symposia,30, 237–249.

    Google Scholar 

  • Fixman, M. (1962). Radius of gyration of polymer chains,Journal of Chemical Physics,36, 306–310.

    Google Scholar 

  • Flory, P. J. (1953).Principles of Polymer Chemistry, Cornell University Press, Ithaca, New York.

    Google Scholar 

  • Flory, P. J. (1969).Statistical Mechanics of Chain Molecules, Wiley, New York.

    Google Scholar 

  • 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.

    Google Scholar 

  • Hsu, L. C. (1948). A theorem on the asymptotic behavior of a multiple integral,Duke Math. J.,15, 623–632.

    Google Scholar 

  • James, A. T. (1964). Distributions of matrix variables and latent roots derived from normal samples,Ann. Math. Statist.,35, 475–501.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Muirhead, R. J. (1978). Latent roots and matrix variates: a review of some asymptotic results,Ann. Statist.,6, 5–33.

    Google Scholar 

  • Muirhead, R. J. (1982).Aspects of Multivariate Statistical Theory, Wiley, New York.

    Google Scholar 

  • Shy, L. Y. and Eichinger, B. E. (1986). Shape distributions for Gaussian molecules: circular and linear chains in two dimensions,Macromolecules,19, 838–843.

    Google Scholar 

  • Stockmayer, W. H. (1974). Statistics of macromolecular shape,XXIVth International Congress of Pure and Applied Chemistry, 91–98, Butterworths, London.

    Google Scholar 

  • Šolc, K. (1971). Shape of a random-flight chain,Journal of Chemical Physics,55, 335–344.

    Google Scholar 

  • Šolc, K. (1972). Statistical mechanics of random-flight chains. III: exact square radii distributions for rings,Macromolecules,5, 705–708.

    Google Scholar 

  • 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.

    Google Scholar 

  • Šolc, K. and Stockmayer, W. H. (1971).Shape of a random-flight chain, Journal of Chemical Physics,54, 2756–2757.

    Google Scholar 

  • Yamakawa, H. (1971).Modern Theory of Polymer Solution, Harper and Row, New York.

    Google Scholar 

  • Wei, G. (1989). Distribution function of the radius of gyration for Gaussian molecules,Journal of Chemical Physics,90, 5873–5877.

    Google Scholar 

  • Wei, G. (1990a). Shape distributions for randomly coiled molecules, Ph.D. dissertation, University of Washington, Seattle.

    Google Scholar 

  • Wei, G. (1990b). A multidimensional integral,SIAM Rev.,32, p. 479.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Wei, G. and Eichinger, B. E. (1990b). Shape distributions for Gaussian molecules: circular and linear chains as asymmetric ellipsoids,Macromolecules,23, 4845–4854.

    Google Scholar 

  • Wei, G. and Eichinger, B. E. (1990c). On shape asymmetry of Gaussian molecules,Journal of Chemical Physics,93, 1430–1435.

    Google Scholar 

  • 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.

    Google Scholar 

  • Zimm, B. H. (1988). Size fluctuations can explain anomalous mobility in field-inverse electrophoresis of DNA,Phys. Rev. Lett.,61, 2965–2968.

    Google Scholar 

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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

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