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11: 31.3 Basic Solutions
The full set of 192 local solutions of (31.2.1), equivalent in 8 sets of 24, resembles Kummer’s set of 24 local solutions of the hypergeometric equation, which are equivalent in 4 sets of 6 solutions (§15.10(ii)); see Maier (2007).
12: 19.4 Derivatives and Differential Equations
If ϕ = π / 2 , then these two equations become hypergeometric differential equations (15.10.1) for K ( k ) and E ( k ) . …
13: 15.19 Methods of Computation
A comprehensive and powerful approach is to integrate the hypergeometric differential equation (15.10.1) by direct numerical methods. …
14: 15.5 Derivatives and Contiguous Functions
15.5.16_5 F ( a , b ; c ; z ) - F ( a - 1 , b ; c ; z ) - ( b / c ) z F ( a , b + 1 ; c + 1 ; z ) = 0 ,
An equivalent equation to the hypergeometric differential equation (15.10.1) is …
15: 16.13 Appell Functions
The following four functions of two real or complex variables x and y cannot be expressed as a product of two F 1 2 functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): …
16: 17.17 Physical Applications
See Kassel (1995). …
17: 32.10 Special Function Solutions
§32.10(vi) Sixth Painlevé Equation
where the fundamental periods 2 ϕ 1 and 2 ϕ 2 are linearly independent functions satisfying the hypergeometric equation
18: 31.11 Expansions in Series of Hypergeometric Functions
§31.11 Expansions in Series of Hypergeometric Functions
§31.11(v) Doubly-Infinite Series
19: Bibliography M
  • R. S. Maier (2005) On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.
  • 20: 31.10 Integral Equations and Representations
    31.10.11 𝒦 ( z , t ) = ( z t - a ) 1 2 - δ - σ ( z t / a ) - 1 2 + δ + σ - α F 1 2 ( 1 2 - δ - σ + α , 3 2 - δ - σ + α - γ α - β + 1 ; a z t ) P { 0 1 0 0 - 1 2 + δ + σ ( z - a ) ( t - a ) ( 1 - a ) ( z t - a ) 1 - ϵ 1 - δ - 1 2 + ϵ - σ } .
    This equation can be solved in terms of hypergeometric functions (§15.11(i)):