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generalized Bessel polynomials

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11: 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 .
12: 10.59 Integrals
§10.59 Integrals
10.59.1 e i b t 𝗃 n ( t ) d t = { π i n P n ( b ) , 1 < b < 1 , 1 2 π ( ± i ) n , b = ± 1 , 0 , ± b > 1 ,
where P n is the Legendre polynomial18.3). For an integral representation of the Dirac delta in terms of a product of spherical Bessel functions of the first kind see §1.17(ii), and for a generalization see Maximon (1991). …
13: 18.12 Generating Functions
18.12.2 𝐅 1 0 ( α + 1 ; ( x 1 ) z 2 ) 𝐅 1 0 ( β + 1 ; ( x + 1 ) z 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 ,
18.12.14 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 .
14: 35.1 Special Notation
Related notations for the Bessel functions are 𝒥 ν + 1 2 ( m + 1 ) ( 𝐓 ) = A ν ( 𝐓 ) / A ν ( 𝟎 ) (Faraut and Korányi (1994, pp. 320–329)), K m ( 0 , , 0 , ν | 𝐒 , 𝐓 ) = | 𝐓 | ν B ν ( 𝐒 𝐓 ) (Terras (1988, pp. 49–64)), and 𝒦 ν ( 𝐓 ) = | 𝐓 | ν B ν ( 𝐒 𝐓 ) (Faraut and Korányi (1994, pp. 357–358)).
15: 16.18 Special Cases
§16.18 Special Cases
The F 1 1 and F 1 2 functions introduced in Chapters 13 and 15, as well as the more general F q p functions introduced in the present chapter, are all special cases of the Meijer G -function. …
16.18.1 F q p ( a 1 , , a p b 1 , , b q ; z ) = ( k = 1 q Γ ( b k ) / k = 1 p Γ ( a k ) ) G p , q + 1 1 , p ( z ; 1 a 1 , , 1 a p 0 , 1 b 1 , , 1 b q ) = ( k = 1 q Γ ( b k ) / k = 1 p Γ ( a k ) ) G q + 1 , p p , 1 ( 1 z ; 1 , b 1 , , b q a 1 , , a p ) .
As a corollary, special cases of the F 1 1 and F 1 2 functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer G -function. …
16: Bibliography C
  • F. Calogero (1978) Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial L n α ( x )  as the index α  and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.
  • CAOP (website) Work Group of Computational Mathematics, University of Kassel, Germany.
  • M. A. Chaudhry and S. M. Zubair (1994) Generalized incomplete gamma functions with applications. J. Comput. Appl. Math. 55 (1), pp. 99–124.
  • P. A. Clarkson and K. Jordaan (2018) Properties of generalized Freud polynomials. J. Approx. Theory 225, pp. 148–175.
  • H. S. Cohl (2013b) On a generalization of the generating function for Gegenbauer polynomials. Integral Transforms Spec. Funct. 24 (10), pp. 807–816.
  • 17: Bibliography Z
  • F. A. Zafiropoulos, T. N. Grapsa, O. Ragos, and M. N. Vrahatis (1996) On the Computation of Zeros of Bessel and Bessel-related Functions. In Proceedings of the Sixth International Colloquium on Differential Equations (Plovdiv, Bulgaria, 1995), D. Bainov (Ed.), Utrecht, pp. 409–416.
  • A. H. Zemanian (1987) Distribution Theory and Transform Analysis, An Introduction and Generalized Functions with Applications. Dover, New York.
  • A. S. Zhedanov (1991) “Hidden symmetry” of Askey-Wilson polynomials. Theoret. and Math. Phys. 89 (2), pp. 1146–1157.
  • A. Zhedanov (1998) On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval. J. Approx. Theory 94 (1), pp. 73–106.
  • Q. Zheng (1997) Generalized Watson Transforms and Applications to Group Representations. Ph.D. Thesis, University of Vermont, Burlington,VT.
  • 18: 8.7 Series Expansions
    8.7.6 Γ ( a , x ) = x a e x n = 0 L n ( a ) ( x ) n + 1 , x > 0 , a < 1 2 .
    19: 10.40 Asymptotic Expansions for Large Argument
    §10.40 Asymptotic Expansions for Large Argument
    §10.40(i) Hankel’s Expansions
    Products
    §10.40(iv) Exponentially-Improved Expansions
    20: 10.23 Sums
    §10.23(i) Multiplication Theorem
    §10.23(iii) Series Expansions of Arbitrary Functions
    For the more general form of expansion …
    Fourier–Bessel Expansion