Gauss%E2%80%93Laguerre%20formula
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1: 18.3 Definitions
§18.3 Definitions
►The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. … ►Table 18.3.1 provides the traditional definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and standardization (§§18.2(i) and 18.2(iii)). … ►For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of for Jacobi polynomials, in powers of for the other cases). … ►Formula (18.3.1) can be understood as a Gauss-Chebyshev quadrature, see (3.5.22), (3.5.23). …2: 15.5 Derivatives and Contiguous Functions
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§15.5(i) Differentiation Formulas
… ►The six functions , , are said to be contiguous to . … ►
15.5.12
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►By repeated applications of (15.5.11)–(15.5.18) any function , in which are integers, can be expressed as a linear combination of and any one of its contiguous functions, with coefficients that are rational functions of , and .
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15.5.20
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3: 16.12 Products
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16.12.1
►The following formula is often referred to as Clausen’s formula
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16.12.2
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16.12.3
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4: 15.4 Special Cases
5: 18.5 Explicit Representations
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►Related formula:
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►For the definitions of , , and see §16.2.
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Laguerre
… ►For corresponding formulas for Chebyshev, Legendre, and the Hermite polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). … ►Laguerre
…6: 15.2 Definitions and Analytical Properties
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§15.2(i) Gauss Series
►The hypergeometric function is defined by the Gauss series … ►On the circle of convergence, , the Gauss series: … ►The same properties hold for , except that as a function of , in general has poles at . … ►Formula (15.4.6) reads . …7: 16.3 Derivatives and Contiguous Functions
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§16.3(i) Differentiation Formulas
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16.3.1
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16.3.3
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►Two generalized hypergeometric functions are (generalized)
contiguous if they have the same pair of values of and , and corresponding parameters differ by integers.
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16.3.6
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8: 3.5 Quadrature
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§3.5(v) Gauss Quadrature
… ►Gauss–Legendre Formula
… ►Gauss–Chebyshev Formula
… ►Gauss–Jacobi Formula
… ►Gauss–Laguerre Formula
…9: 15.8 Transformations of Variable
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►The transformation formulas between two hypergeometric functions in Group 2, or two hypergeometric functions in Group 3, are the linear transformations (15.8.1).
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15.8.13
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15.8.14
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15.8.15
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15.8.16
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10: 16.4 Argument Unity
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►The function is well-poised if
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►The function with argument unity and general values of the parameters is discussed in Bühring (1992).
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►For generalizations involving functions see Kim et al. (2013).
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►Balanced series have transformation formulas and three-term relations.
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►Transformations for both balanced and very well-poised are included in Bailey (1964, pp. 56–63).
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