Laguere EOP’s
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11: 18.39 Applications in the Physical Sciences
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►While non-normalizable continuum, or scattering, states are mentioned, with appropriate references in what follows, focus is on the eigenfunctions corresponding to the point, or discrete, spectrum, and representing bound rather than scattering states, these former being expressed in terms of OP’s or EOP’s.
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►c) A Rational SUSY Potential
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►Thus the two missing quantum numbers corresponding to EOP’s of order and of the type III Hermite EOP’s are offset in the node counts by the fact that all excited state eigenfunctions also have two missing real zeros.
Kuijlaars and Milson (2015, §1) refer to these, in this case complex zeros, as exceptional, as opposed to regular, zeros of the EOP’s, these latter belonging to the (real) orthogonality integration range.
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►These are overviewed in §18.38(iii), and §18.36(vi), and typically involve OP’s or EOP’s.
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12: 18.1 Notation
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-Hahn Class OP’s
… ►Askey–Wilson Class OP’s
… ►Associated OP’s
… ►Other OP’s
… ►Classical OP’s in Two Variables
…13: 31.13 Asymptotic Approximations
§31.13 Asymptotic Approximations
… ►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). ►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).14: 28.16 Asymptotic Expansions for Large
15: 25.4 Reflection Formulas
16: 25.13 Periodic Zeta Function
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►The notation is used for the polylogarithm with real:
…where if is an integer, otherwise.
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is periodic in with period 1, and equals when is an integer.
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25.13.2
, ,
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25.13.3
if ; if .
17: 27.4 Euler Products and Dirichlet Series
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►The completely multiplicative function gives the Euler product representation of the Riemann zeta function
(§25.2(i)):
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►The Riemann zeta function is the prototype of series of the form
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27.4.4
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►The function is a generating function, or more precisely, a Dirichlet generating
function, for the coefficients.
…In (27.4.12) and (27.4.13) is the derivative of .
18: 25.1 Special Notation
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►The main function treated in this chapter is the Riemann zeta function .
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►The main related functions are the Hurwitz zeta function , the dilogarithm , the polylogarithm (also known as Jonquière’s function ), Lerch’s transcendent , and the Dirichlet -functions .
nonnegative integers. | |
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complex variable. | |
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Euler’s constant (§5.2(ii)). | |
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19: 25.20 Approximations
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Cody et al. (1971) gives rational approximations for in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are , , , . Precision is varied, with a maximum of 20S.
Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of and , , for (23D).