Gauss%E2%80%93Legendre%20formula
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1: 14.1 Special Notation
§14.1 Special Notation
… βΊThe main functions treated in this chapter are the Legendre functions , , , ; Ferrers functions , (also known as the Legendre functions on the cut); associated Legendre functions , , ; conical functions , , , , (also known as Mehler functions). … βΊMagnus et al. (1966) denotes , , , and by , , , and , respectively. Hobson (1931) denotes both and by ; similarly for and .2: 18.3 Definitions
§18.3 Definitions
… βΊThis table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … βΊFormula (18.3.1) can be understood as a Gauss-Chebyshev quadrature, see (3.5.22), (3.5.23). … βΊLegendre
βΊLegendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …3: 15.5 Derivatives and Contiguous Functions
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§15.5(i) Differentiation Formulas
… βΊThe six functions , , are said to be contiguous to . … βΊ
15.5.12
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βΊBy 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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4: 16.12 Products
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16.12.1
βΊThe 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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5: 15.4 Special Cases
6: 15.16 Products
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15.16.1
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15.16.2
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15.16.3
, , .
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15.16.4
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Generalized Legendre’s Relation
…7: 15.2 Definitions and Analytical Properties
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§15.2(i) Gauss Series
βΊThe hypergeometric function is defined by the Gauss series … βΊOn the circle of convergence, , the Gauss series: … βΊThe same properties hold for , except that as a function of , in general has poles at . … βΊFormula (15.4.6) reads . …8: 18.5 Explicit Representations
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§18.5(ii) Rodrigues Formulas
… βΊRelated formula: … βΊFor the definitions of , , and see §16.2. … βΊFor corresponding formulas for Chebyshev, Legendre, and the Hermite polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). … βΊLegendre
…9: 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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βΊTwo 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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