generalized hypergeometric differential equation
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11: Bibliography B
12: Bibliography O
§15.17(i) Differential Equations… ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … … ►Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …
14: Bibliography N
§35.7 Gaussian Hypergeometric Function of Matrix Argument… ►
§35.7(iii) Partial Differential Equations… ►Subject to the conditions (a)–(c), the function is the unique solution of each partial differential equation … ►Systems of partial differential equations for the (defined in §35.8) and functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). …
16: Bibliography M
§32.10(iv) Fourth Painlevé Equation… ►
§32.10(vi) Sixth Painlevé Equation… ►where the fundamental periods and are linearly independent functions satisfying the hypergeometric equation … ►
§19.16(ii)►All 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 … ►For generalizations and further information, especially representation of the -function as a Dirichlet average, see Carlson (1977b). ►
§19.16(iii) Various Cases of…
In Equation (1.13.4), the determinant form of the two-argument Wronskian
was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the -argument Wronskian is given by , where . Immediately below Equation (1.13.4), a sentence was added giving the definition of the -argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for th-order differential equations. A reference to Ince (1926, §5.2) was added.
These equations have been generalized to include the additional cases of , , respectively.
There were clarifications made in the conditions on the parameter in of those equations.
The Wronskian was generalized to include both associated Legendre and Ferrers functions.
has been generalized to cover an additional case.