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11: 35.4 Partitions and Zonal Polynomials
12: 2.6 Distributional Methods
On inserting this identity into (2.6.54), we immediately encounter divergent integrals of the form …
13: 15.17 Mathematical Applications
§15.17(ii) Conformal Mappings
Hypergeometric functions, especially complete elliptic integrals, also play an important role in quasiconformal mapping. …
§15.17(iv) Combinatorics
In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. … Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). …
14: 35.6 Confluent Hypergeometric Functions of Matrix Argument
35.6.2 Ψ ( a ; b ; 𝐓 ) = 1 Γ m ( a ) 𝛀 etr ( 𝐓 𝐗 ) | 𝐗 | a 1 2 ( m + 1 ) | 𝐈 + 𝐗 | b a 1 2 ( m + 1 ) d 𝐗 , ( a ) > 1 2 ( m 1 ) , 𝐓 𝛀 .
15: 19.7 Connection Formulas
§19.7 Connection Formulas
§19.7(i) Complete Integrals of the First and Second Kinds
Reciprocal-Modulus Transformation
Provided the functions in these identities are correctly analytically continued in the complex β -plane, then the identities will also hold in the complex β -plane.
Imaginary-Modulus Transformation
16: 22.9 Cyclic Identities
§22.9 Cyclic Identities
§22.9(ii) Typical Identities of Rank 2
§22.9(iii) Typical Identities of Rank 3
17: 21.7 Riemann Surfaces
§21.7(ii) Fay’s Trisecant Identity
where again all integration paths are identical for all components. Generalizations of this identity are given in Fay (1973, Chapter 2). …
§21.7(iii) Frobenius’ Identity
18: 6.12 Asymptotic Expansions
§6.12(i) Exponential and Logarithmic Integrals
For the function χ see §9.7(i). …
§6.12(ii) Sine and Cosine Integrals
19: 35.8 Generalized Hypergeometric Functions of Matrix Argument
35.8.13 𝟎 < 𝐗 < 𝐈 | 𝐗 | a 1 1 2 ( m + 1 ) | 𝐈 𝐗 | b 1 a 1 1 2 ( m + 1 ) F q p ( a 2 , , a p + 1 b 2 , , b q + 1 ; 𝐓 𝐗 ) d 𝐗 = 1 B m ( b 1 a 1 , a 1 ) F q + 1 p + 1 ( a 1 , , a p + 1 b 1 , , b q + 1 ; 𝐓 ) , ( b 1 a 1 ) , ( a 1 ) > 1 2 ( m 1 ) .
20: 35.7 Gaussian Hypergeometric Function of Matrix Argument