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expansion in zonal polynomials

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1: 35.8 Generalized Hypergeometric Functions of Matrix Argument
β–ΊThe generalized hypergeometric function F q p with matrix argument 𝐓 𝓒 , numerator parameters a 1 , , a p , and denominator parameters b 1 , , b q is …
2: 35.10 Methods of Computation
β–ΊFor small values of β€– 𝐓 β€– the zonal polynomial expansion given by (35.8.1) can be summed numerically. For large β€– 𝐓 β€– the asymptotic approximations referred to in §35.7(iv) are available. … β–ΊSee Yan (1992) for the F 1 1 and F 1 2 functions of matrix argument in the case m = 2 , and Bingham et al. (1992) for Monte Carlo simulation on 𝐎 ⁑ ( m ) applied to a generalization of the integral (35.5.8). β–ΊKoev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). These algorithms are extremely efficient, converge rapidly even for large values of m , and have complexity linear in m .
3: Bibliography L
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  • L. Lapointe and L. Vinet (1996) Exact operator solution of the Calogero-Sutherland model. Comm. Math. Phys. 178 (2), pp. 425–452.
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  • X. Li and R. Wong (2000) A uniform asymptotic expansion for Krawtchouk polynomials. J. Approx. Theory 106 (1), pp. 155–184.
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  • J. L. López and N. M. Temme (1999a) Approximation of orthogonal polynomials in terms of Hermite polynomials. Methods Appl. Anal. 6 (2), pp. 131–146.
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  • J. L. López and N. M. Temme (2010a) Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions. Numer. Math. 116 (2), pp. 269–289.
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  • Y. L. Luke and J. Wimp (1963) Jacobi polynomial expansions of a generalized hypergeometric function over a semi-infinite ray. Math. Comp. 17 (84), pp. 395–404.
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  • R. B. Paris (2001a) On the use of Hadamard expansions in hyperasymptotic evaluation. I. Real variables. Proc. Roy. Soc. London Ser. A 457 (2016), pp. 2835–2853.
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  • R. B. Paris (2001b) On the use of Hadamard expansions in hyperasymptotic evaluation. II. Complex variables. Proc. Roy. Soc. London Ser. A 457, pp. 2855–2869.
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  • R. B. Paris (2002a) Error bounds for the uniform asymptotic expansion of the incomplete gamma function. J. Comput. Appl. Math. 147 (1), pp. 215–231.
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  • A. M. Parkhurst and A. T. James (1974) Zonal Polynomials of Order 1 Through 12 . In Selected Tables in Mathematical Statistics, H. L. Harter and D. B. Owen (Eds.), Vol. 2, pp. 199–388.
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  • R. Piessens and M. Branders (1972) Chebyshev polynomial expansions of the Riemann zeta function. Math. Comp. 26 (120), pp. G1–G5.
  • 5: 14.18 Sums
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    §14.18(i) Expansion Theorem
    β–ΊFor expansions of arbitrary functions in series of Legendre polynomials see §18.18(i), and for expansions of arbitrary functions in series of associated Legendre functions see Schäfke (1961b). … β–Ί
    Zonal Harmonic Series
    β–ΊFor a series representation of the Dirac delta in terms of products of Legendre polynomials see (1.17.22). …
    6: Bibliography G
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  • B. Gabutti and B. Minetti (1981) A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function. J. Comput. Phys. 42 (2), pp. 277–287.
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  • W. Gautschi (1984) Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.
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  • W. Gautschi (1996) Orthogonal Polynomials: Applications and Computation. In Acta Numerica, 1996, A. Iserles (Ed.), Acta Numerica, Vol. 5, pp. 45–119.
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  • K. I. Gross and D. St. P. Richards (1987) Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions. Trans. Amer. Math. Soc. 301 (2), pp. 781–811.
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  • E. Grosswald (1978) Bessel Polynomials. Lecture Notes in Mathematics, Vol. 698, Springer, Berlin-New York.
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  • A. Takemura (1984) Zonal Polynomials. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 4, Institute of Mathematical Statistics, Hayward, CA.
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  • N. M. Temme (1978) Uniform asymptotic expansions of confluent hypergeometric functions. J. Inst. Math. Appl. 22 (2), pp. 215–223.
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  • N. M. Temme (1979b) The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10 (4), pp. 757–766.
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  • N. M. Temme (1990b) Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. SIAM J. Math. Anal. 21 (1), pp. 241–261.
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  • E. C. Titchmarsh (1946) Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.
  • 8: 18.38 Mathematical Applications
    β–ΊIn consequence, expansions of functions that are infinitely differentiable on [ 1 , 1 ] in series of Chebyshev polynomials usually converge extremely rapidly. … β–Ί
    Zonal Spherical Harmonics
    β–ΊUltraspherical polynomials are zonal spherical harmonics. … β–ΊHermite polynomials (and their Freud-weight analogs (§18.32)) play an important role in random matrix theory. … β–ΊDunkl type operators and nonsymmetric polynomials have been associated with various other families in the Askey scheme and q -Askey scheme, in particular with Wilson polynomials, see Groenevelt (2007), and with Jacobi polynomials, see Koornwinder and Bouzeffour (2011, §7). …
    9: Bibliography S
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  • H. Shanker (1939) On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.
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  • A. Sidi (2011) Asymptotic expansion of Mellin transforms in the complex plane. Int. J. Pure Appl. Math. 71 (3), pp. 465–480.
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  • J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.
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  • Stembridge (website) John Stembridge’s Home Page
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  • W. F. Sun (1996) Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.