hypergeometric differential equation
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21: Bibliography L
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Some Differential Equations Satisfied by Hypergeometric Functions.
In Approximation and Computation (West Lafayette, IN, 1993),
Internat. Ser. Numer. Math., Vol. 119, pp. 371–381.
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22: Bibliography M
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On reducing the Heun equation to the hypergeometric equation.
J. Differential Equations 213 (1), pp. 171–203.
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23: Frank W. J. Olver
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►He is particularly known for his extensive work in the study of the asymptotic solution of differential equations, i.
…, the behavior of solutions as the independent variable, or some parameter, tends to infinity, and in the study of the particular solutions of differential equations known as special functions (e.
…, Bessel functions, hypergeometric functions, Legendre functions).
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24: 13.27 Mathematical Applications
§13.27 Mathematical Applications
►Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. … ►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i).25: 12.18 Methods of Computation
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►Because PCFs are special cases of confluent hypergeometric functions, the methods of computation described in §13.29 are applicable to PCFs.
These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions.
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26: 13.3 Recurrence Relations and Derivatives
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13.3.14
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27: 31.12 Confluent Forms of Heun’s Equation
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►Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity.
This is analogous to the derivation of the confluent hypergeometric equation from the hypergeometric equation in §13.2(i).
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Confluent Heun Equation
… ►Biconfluent Heun Equation
… ►Triconfluent Heun Equation
…28: 17.13 Integrals
29: 16.2 Definition and Analytic Properties
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§16.2(i) Generalized Hypergeometric Series
… ► … ►Polynomials
… ►Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via … ►§16.2(v) Behavior with Respect to Parameters
…30: 16.14 Partial Differential Equations
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►In addition to the four Appell functions there are other sums of double series that cannot be expressed as a product of two functions, and which satisfy pairs of linear partial differential equations of the second order.
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