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1: 15.2 Definitions and Analytical Properties
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§15.2(i) Gauss Series
โบThe hypergeometric function is defined by the Gauss series … … โบ§15.2(ii) Analytic Properties
… โบThe same properties hold for , except that as a function of , in general has poles at . …2: 16.2 Definition and Analytic Properties
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§16.2(i) Generalized Hypergeometric Series
… โบ … โบPolynomials
… โบNote also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via … โบ§16.2(v) Behavior with Respect to Parameters
…3: 17.1 Special Notation
§17.1 Special Notation
… โบnonnegative integers. | |
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4: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
โบ§35.8(i) Definition
… โบConvergence Properties
… โบConfluence
… โบInvariance
…5: 35.6 Confluent Hypergeometric Functions of Matrix Argument
§35.6 Confluent Hypergeometric Functions of Matrix Argument
โบ§35.6(i) Definitions
… โบLaguerre Form
… โบ§35.6(ii) Properties
… โบ§35.6(iii) Relations to Bessel Functions of Matrix Argument
…6: 35.7 Gaussian Hypergeometric Function of Matrix Argument
§35.7 Gaussian Hypergeometric Function of Matrix Argument
โบ§35.7(i) Definition
… โบJacobi Form
… โบConfluent Form
… โบIntegral Representation
…7: 15.10 Hypergeometric Differential Equation
§15.10 Hypergeometric Differential Equation
… โบSingularity
… โบSingularity
… โบSingularity
… โบThe connection formulas for the principal branches of Kummer’s solutions are: …8: 19.16 Definitions
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§19.16(ii)
โบAll elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function …The -function is often used to make a unified statement of a property of several elliptic integrals. … โบ โบ§19.16(iii) Various Cases of
…9: 35.1 Special Notation
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โบ(For other notation see Notation for the Special Functions.)
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โบThe main functions treated in this chapter are the multivariate gamma and beta functions, respectively and , and the special functions of matrix argument: Bessel (of the first kind) and (of the second kind) ; confluent hypergeometric (of the first kind) or and (of the second kind) ; Gaussian hypergeometric
or ; generalized hypergeometric
or .
โบAn alternative notation for the multivariate gamma function is (Herz (1955, p. 480)).
Related notations for the Bessel functions are (Faraut and Korányi (1994, pp. 320–329)), (Terras (1988, pp. 49–64)), and (Faraut and Korányi (1994, pp. 357–358)).
complex variables. | |
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