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1: 18.34 Bessel Polynomials
§18.34(i) Definitions and Recurrence Relation
Often only the polynomials (18.34.2) are called Bessel polynomials, while the polynomials (18.34.1) and (18.34.3) are called generalized Bessel polynomials. See also §10.49(ii). …
2: Bibliography D
  • M. G. de Bruin, E. B. Saff, and R. S. Varga (1981a) On the zeros of generalized Bessel polynomials. I. Nederl. Akad. Wetensch. Indag. Math. 84 (1), pp. 1–13.
  • M. G. de Bruin, E. B. Saff, and R. S. Varga (1981b) On the zeros of generalized Bessel polynomials. II. Nederl. Akad. Wetensch. Indag. Math. 84 (1), pp. 14–25.
  • T. M. Dunster (2001c) Uniform asymptotic expansions for the reverse generalized Bessel polynomials, and related functions. SIAM J. Math. Anal. 32 (5), pp. 987–1013.
  • 3: Bibliography W
  • R. Wong and J.-M. Zhang (1997) Asymptotic expansions of the generalized Bessel polynomials. J. Comput. Appl. Math. 85 (1), pp. 87–112.
  • 4: 18.15 Asymptotic Approximations
    18.15.19 L n ( α ) ( ν x ) = e 1 2 ν x 2 α x 1 2 α + 1 4 ( 1 - x ) 1 4 ( ξ 1 2 J α ( ν ξ ) m = 0 M - 1 A m ( ξ ) ν 2 m + ξ - 1 2 J α + 1 ( ν ξ ) m = 0 M - 1 B m ( ξ ) ν 2 m + 1 + ξ 1 2 env J α ( ν ξ ) O ( 1 ν 2 M - 1 ) ) ,
    5: Bibliography L
  • J. L. López and N. M. Temme (1999b) Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. J. Math. Anal. Appl. 239 (2), pp. 457–477.
  • 6: 35.9 Applications
    §35.9 Applications
    In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument F q p , with p 2 and q 1 . … These references all use results related to the integral formulas (35.4.7) and (35.5.8). … In chemistry, Wei and Eichinger (1993) expresses the probability density functions of macromolecules in terms of generalized hypergeometric functions of matrix argument, and develop asymptotic approximations for these density functions. In the nascent area of applications of zonal polynomials to the limiting probability distributions of symmetric random matrices, one of the most comprehensive accounts is Rains (1998).
    7: 10.49 Explicit Formulas
    §10.49(i) Unmodified Functions
    §10.49(ii) Modified Functions
    k = 0 n a k ( n + 1 2 ) z n - k is sometimes called the Bessel polynomial of degree n . For a survey of properties of these polynomials and their generalizations see Grosswald (1978). … …
    8: 18.11 Relations to Other Functions
    18.11.6 lim n 1 n α L n ( α ) ( z n ) = 1 z 1 2 α J α ( 2 z 1 2 ) .
    9: 8.7 Series Expansions
    8.7.6 Γ ( a , x ) = x a e - x n = 0 L n ( a ) ( x ) n + 1 , x > 0 .
    10: 18.10 Integral Representations
    18.10.9 L n ( α ) ( x ) = e x x - 1 2 α n ! 0 e - t t n + 1 2 α J α ( 2 x t ) d t , α > - 1 .