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asymptotic approximations and expansions

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1: 14.26 Uniform Asymptotic Expansions
§14.26 Uniform Asymptotic Expansions
2: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). …
3: 18.24 Hahn Class: Asymptotic Approximations
§18.24 Hahn Class: Asymptotic Approximations
Asymptotic approximations are also provided for the zeros of K n ( x ; p , N ) in various cases depending on the values of p and μ . … For asymptotic approximations for the zeros of M n ( n x ; β , c ) in terms of zeros of Ai ( x ) 9.9(i)), see Jin and Wong (1999) and Khwaja and Olde Daalhuis (2012). …
Approximations in Terms of Laguerre Polynomials
Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.
4: 18.32 OP’s with Respect to Freud Weights
However, for asymptotic approximations in terms of elementary functions for the OP’s, and also for their largest zeros, see Levin and Lubinsky (2001) and Nevai (1986). For a uniform asymptotic expansion in terms of Airy functions (§9.2) for the OP’s in the case Q ( x ) = x 4 see Bo and Wong (1999). For asymptotic approximations to OP’s that correspond to Freud weights with more general functions Q ( x ) see Deift et al. (1999a, b), Bleher and Its (1999), and Kriecherbauer and McLaughlin (1999).
5: 28.26 Asymptotic Approximations for Large q
§28.26 Asymptotic Approximations for Large q
§28.26(i) Goldstein’s Expansions
The asymptotic expansions of Fs m ( z , h ) and Gs m ( z , h ) in the same circumstances are also given by the right-hand sides of (28.26.4) and (28.26.5), respectively. …
§28.26(ii) Uniform Approximations
For asymptotic approximations for M ν ( 3 , 4 ) ( z , h ) see also Naylor (1984, 1987, 1989).
6: 2.1 Definitions and Elementary Properties
means that for each n , the difference between f ( x ) and the n th partial sum on the right-hand side is O ( ( x - c ) n ) as x c in X . … Some asymptotic approximations are expressed in terms of two or more Poincaré asymptotic expansions. …For an example see (2.8.15). …
§2.1(iv) Uniform Asymptotic Expansions
§2.1(v) Generalized Asymptotic Expansions
7: 10.70 Zeros
§10.70 Zeros
Asymptotic approximations for large zeros are as follows. …
zeros of  ber ν x 2 ( t - f ( t ) ) , t = ( m - 1 2 ν - 3 8 ) π ,
zeros of  bei ν x 2 ( t - f ( t ) ) , t = ( m - 1 2 ν + 1 8 ) π ,
zeros of  ker ν x 2 ( t + f ( - t ) ) , t = ( m - 1 2 ν - 5 8 ) π ,
8: 15.12 Asymptotic Approximations
§15.12 Asymptotic Approximations
§15.12(i) Large Variable
§15.12(ii) Large c
For this result and an extension to an asymptotic expansion with error bounds see Jones (2001). … For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).
9: 33.21 Asymptotic Approximations for Large | r |
§33.21(i) Limiting Forms
§33.21(ii) Asymptotic Expansions
10: 29.16 Asymptotic Expansions
§29.16 Asymptotic Expansions
Hargrave and Sleeman (1977) give asymptotic approximations for Lamé polynomials and their eigenvalues, including error bounds. The approximations for Lamé polynomials hold uniformly on the rectangle 0 z K , 0 z K , when n k and n k assume large real values. The approximating functions are exponential, trigonometric, and parabolic cylinder functions.