Jacobi polynomials
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11: 18.37 Classical OP’s in Two or More Variables
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Definition in Terms of Jacobi Polynomials
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18.37.1
, , .
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Definition in Terms of Jacobi Polynomials
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18.37.7
, .
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►In several variables they occur, for , as Jack polynomials and also as Jacobi polynomials associated with
root systems; see Macdonald (1995, Chapter VI, §10), Stanley (1989), Kuznetsov and Sahi (2006, Part 1), Heckman (1991).
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12: 18.18 Sums
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18.18.2
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►See Andrews et al. (1999, Lemma 7.1.1) for the more general expansion of in terms of .
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Jacobi
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18.18.25
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18.18.26
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13: 18.11 Relations to Other Functions
14: 31.16 Mathematical Applications
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►Expansions of Heun polynomial products in terms of Jacobi polynomial (§18.3) products are derived in Kalnins and Miller (1991a, b, 1993) from the viewpoint of interrelation between two bases in a Hilbert space:
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31.16.1
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15: 18.27 -Hahn Class
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18.27.10
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18.27.12_5
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18.27.14_2
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18.27.14_4
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►Little -Jacobi polynomials
for are called little -Laguerre or Wall polynomials:
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16: 18.17 Integrals
17: 18.15 Asymptotic Approximations
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§18.15(i) Jacobi
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18.15.1
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►For large , fixed , and , Dunster (1999) gives asymptotic expansions of that are uniform in unbounded complex -domains containing .
…This reference also supplies asymptotic expansions of for large , fixed , and .
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►For an asymptotic expansion of as that holds uniformly for complex bounded away from , see Elliott (1971).
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18: 18.34 Bessel Polynomials
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18.34.8
►In this limit the finite system of Jacobi polynomials
which is orthogonal on (see §18.3) tends to the finite system of Romanovski–Bessel polynomials which is orthogonal on (see (18.34.5_5)).
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19: 18.21 Hahn Class: Interrelations
20: 18.5 Explicit Representations
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18.5.7
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18.5.8
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►The first of each of equations (18.5.7) and (18.5.8) can be regarded as definitions of when the conditions and are not satisfied.
…For this reason, and also in the interest of simplicity, in the case of the Jacobi polynomials
we assume throughout this chapter that and , unless stated otherwise.
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