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generalized hypergeometric differential equation

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1: 16.8 Differential Equations
§16.8(ii) The Generalized Hypergeometric Differential Equation
When no b j is an integer, and no two b j differ by an integer, a fundamental set of solutions of (16.8.3) is given by … We have the connection formula …
§16.8(iii) Confluence of Singularities
2: 15.11 Riemann’s Differential Equation
15.11.3 w = P { α β γ a 1 b 1 c 1 z a 2 b 2 c 2 } .
The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by …
15.11.8 z λ ( 1 - z ) μ P { 0 1 a 1 b 1 c 1 z a 2 b 2 c 2 } = P { 0 1 a 1 + λ b 1 + μ c 1 - λ - μ z a 2 + λ b 2 + μ c 2 - λ - μ } ,
3: 16.23 Mathematical Applications
§16.23 Mathematical Applications
§16.23(i) Differential Equations
These equations are frequently solvable in terms of generalized hypergeometric functions, and the monodromy of generalized hypergeometric functions plays an important role in describing properties of the solutions. … …
§16.23(iv) Combinatorics and Number Theory
4: 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 } .
5: 31.11 Expansions in Series of Hypergeometric Functions
6: Howard S. Cohl
Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and q -series. …
7: 16.25 Methods of Computation
§16.25 Methods of Computation
Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. They are similar to those described for confluent hypergeometric functions, and hypergeometric functions in §§13.29 and 15.19. There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …
8: 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): …
9: 16.2 Definition and Analytic Properties
§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
10: Bibliography L
  • C. G. Lambe and D. R. Ward (1934) Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.
  • E. W. Leaver (1986) Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.
  • J. Letessier, G. Valent, and J. Wimp (1994) Some Differential Equations Satisfied by Hypergeometric Functions. In Approximation and Computation (West Lafayette, IN, 1993), Internat. Ser. Numer. Math., Vol. 119, pp. 371–381.
  • N. A. Lukaševič (1971) The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).
  • Y. L. Luke and J. Wimp (1963) Jacobi polynomial expansions of a generalized hypergeometric function over a semi-infinite ray. Math. Comp. 17 (84), pp. 395–404.