Gauss transformations
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11: 31.7 Relations to Other Functions
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§31.7(i) Reductions to the Gauss Hypergeometric Function
… โบOther reductions of to a , with at least one free parameter, exist iff the pair takes one of a finite number of values, where . …They are analogous to quadratic and cubic hypergeometric transformations (§§15.8(iii)–15.8(v)). … โบJoyce (1994) gives a reduction in which the independent variable is transformed not polynomially or rationally, but algebraically. … โบThe solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …12: 35.7 Gaussian Hypergeometric Function of Matrix Argument
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Transformations of Parameters
… โบGauss Formula
… โบSubject to the conditions (a)–(c), the function is the unique solution of each partial differential equation … โบSystems of partial differential equations for the (defined in §35.8) and functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). … โบ
35.7.10
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13: 16.24 Physical Applications
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โบThey can be expressed as functions with unit argument.
The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner symbols.
These are balanced functions with unit argument.
Lastly, special cases of the symbols are functions with unit argument.
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14: 13.23 Integrals
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13.23.1
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15: 3.5 Quadrature
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§3.5(v) Gauss Quadrature
… โบGauss–Legendre Formula
… โบGauss–Chebyshev Formula
… โบGauss–Jacobi Formula
… โบGauss–Laguerre Formula
…16: 18.17 Integrals
17: 15.8 Transformations of Variable
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15.8.28
, .
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18: 15.19 Methods of Computation
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โบThe Gauss series (15.2.1) converges for .
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โบLarge values of or , for example, delay convergence of the Gauss series, and may also lead to severe cancellation.
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โบGauss quadrature approximations are discussed in Gautschi (2002b).
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โบFor example, in the half-plane we can use (15.12.2) or (15.12.3) to compute and , where is a large positive integer, and then apply (15.5.18) in the backward direction.
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19: 15.12 Asymptotic Approximations
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โบBy combination of the foregoing results of this subsection with the linear transformations of §15.8(i) and the connection formulas of §15.10(ii), similar asymptotic approximations for can be obtained with or , .
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