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1: 18.34 Bessel Polynomials
§18.34 Bessel Polynomials
See also §10.49(ii). …
§18.34(ii) Orthogonality
18.34.7 x 2 y n ′′ ( x ; a ) + ( a x + 2 ) y n ( x ; a ) - n ( n + a - 1 ) y n ( x ; a ) = 0 ,
2: 18.1 Notation
  • Bessel: y n ( x ; a ) .

  • 3: 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.
  • 4: 10.49 Explicit Formulas
    10.49.8 i n ( 1 ) ( z ) = 1 2 e z k = 0 n ( - 1 ) k a k ( n + 1 2 ) z k + 1 + ( - 1 ) n + 1 1 2 e - z k = 0 n a k ( n + 1 2 ) z k + 1 .
    10.49.10 i n ( 2 ) ( z ) = 1 2 e z k = 0 n ( - 1 ) k a k ( n + 1 2 ) z k + 1 + ( - 1 ) n 1 2 e - z k = 0 n a k ( n + 1 2 ) z k + 1 .
    k = 0 n a k ( n + 1 2 ) z n - k is sometimes called the Bessel polynomial of degree n . … …
    5: Bibliography E
  • Á. Elbert (2001) Some recent results on the zeros of Bessel functions and orthogonal polynomials. J. Comput. Appl. Math. 133 (1-2), pp. 65–83.
  • W. D. Evans, W. N. Everitt, K. H. Kwon, and L. L. Littlejohn (1993) Real orthogonalizing weights for Bessel polynomials. J. Comput. Appl. Math. 49 (1-3), pp. 51–57.
  • 6: 10.23 Sums
    10.23.8 𝒞 ν ( w ) w ν = 2 ν Γ ( ν ) k = 0 ( ν + k ) 𝒞 ν + k ( u ) u ν J ν + k ( v ) v ν C k ( ν ) ( cos α ) , ν 0 , - 1 , , | v e ± i α | < | u | ,
    10.23.9 e i v cos α = Γ ( ν ) ( 1 2 v ) ν k = 0 ( ν + k ) i k J ν + k ( v ) C k ( ν ) ( cos α ) , ν 0 , - 1 , .
    10.23.12 1 t - z = J 0 ( z ) O 0 ( t ) + 2 k = 1 J k ( z ) O k ( t ) , | z | < | t | .
    7: 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.
  • 8: 18.12 Generating Functions
    18.12.2 ( 1 2 ( 1 - x ) z ) - 1 2 α J α ( 2 ( 1 - x ) z ) ( 1 2 ( 1 + x ) z ) - 1 2 β I β ( 2 ( 1 + x ) z ) = n = 0 P n ( α , β ) ( x ) Γ ( n + α + 1 ) Γ ( n + β + 1 ) z n .
    18.12.6 Γ ( λ + 1 2 ) e z cos θ ( 1 2 z sin θ ) 1 2 - λ J λ - 1 2 ( z sin θ ) = n = 0 C n ( λ ) ( cos θ ) ( 2 λ ) n z n , 0 θ π .
    18.12.12 e x z J 0 ( z 1 - x 2 ) = n = 0 P n ( x ) n ! z n .
    18.12.14 Γ ( α + 1 ) ( x z ) - 1 2 α e z J α ( 2 x z ) = n = 0 L n ( α ) ( x ) ( α + 1 ) n z n .
    9: 3.5 Quadrature
    are related to Bessel polynomials (§§10.49(ii) and 18.34). … …
    10: Bibliography G
  • E. Grosswald (1978) Bessel Polynomials. Lecture Notes in Mathematics, Vol. 698, Springer, Berlin-New York.