with respect to degree or order
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31—40 of 290 matching pages
31: 2.8 Differential Equations with a Parameter
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►For example, can be the order of a Bessel function or degree of an orthogonal polynomial.
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§2.8(iv) Case III: Simple Pole
… ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). … ►For a coalescing turning point and double pole see Boyd and Dunster (1986) and Dunster (1990b); in this case the uniform approximants are Bessel functions of variable order. … ►Lastly, for an example of a fourth-order differential equation, see Wong and Zhang (2007). …32: 18.33 Polynomials Orthogonal on the Unit Circle
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►A system of polynomials , , where is of proper degree
, is orthonormal on the unit circle with respect
to the weight function
() if
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►A system of monic polynomials , , where is of proper degree
, is orthogonal on the unit circle with respect
to the measure
if
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33: Bibliography O
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Hyperasymptotic solutions of second-order linear differential equations. I.
Methods Appl. Anal. 2 (2), pp. 173–197.
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On the calculation of Stokes multipliers for linear differential equations of the second order.
Methods Appl. Anal. 2 (3), pp. 348–367.
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Hyperasymptotic solutions of second-order linear differential equations. II.
Methods Appl. Anal. 2 (2), pp. 198–211.
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Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one.
Proc. Roy. Soc. London Ser. A 454, pp. 1–29.
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Second-order differential equations with fractional transition points.
Trans. Amer. Math. Soc. 226, pp. 227–241.
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34: 24.16 Generalizations
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►
§24.16(i) Higher-Order Analogs
►Polynomials and Numbers of Integer Order
►For , Bernoulli and Euler polynomials of order are defined respectively by …When they reduce to the Bernoulli and Euler numbers of order : … ►In no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989)); -adic integer order Bernoulli numbers (Adelberg (1996)); -adic -Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli–Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).35: 14.26 Uniform Asymptotic Expansions
§14.26 Uniform Asymptotic Expansions
…36: 14.7 Integer Degree and Order
§14.7 Integer Degree and Order
… ►37: 10.57 Uniform Asymptotic Expansions for Large Order
§10.57 Uniform Asymptotic Expansions for Large Order
…38: Frank W. J. Olver
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►Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history.
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39: 10.72 Mathematical Applications
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►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter.
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►In regions in which (10.72.1) has a simple turning point , that is, and are analytic (or with weaker conditions if is a real variable) and is a simple zero of , asymptotic expansions of the solutions for large can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order
(§9.6(i)).
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►If has a double zero , or more generally is a zero of order
, , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order
.
…The order of the approximating Bessel functions, or modified Bessel functions, is , except in the case when has a double pole at .
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►Then for large asymptotic approximations of the solutions can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on and ).
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40: 23.20 Mathematical Applications
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►Let denote the set of points on that are of finite order (that is, those points for which there exists a positive integer with ), and let be the sets of points with integer and rational coordinates, respectively.
…The resulting points are then tested for finite order as follows.
…If any of these quantities is zero, then the point has finite order.
If any of , , is not an integer, then the point has infinite order.
…If none of these equalities hold, then has infinite order.
…