# generalized hypergeometric differential equation

(0.008 seconds)

## 11—20 of 49 matching pages

##### 11: Bibliography B

##### 12: Bibliography O

##### 13: 15.17 Mathematical Applications

###### §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

##### 15: 35.7 Gaussian Hypergeometric Function of Matrix Argument

###### §35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►###### §35.7(iii) Partial Differential Equations

… ►Subject to the conditions (a)–(c), the function $f(\mathbf{T})={}_{2}F_{1}(a,b;c;\mathbf{T})$ is the unique solution of each partial differential equation … ►Systems of partial differential equations for the ${}_{0}F_{1}$ (defined in §35.8) and ${}_{1}F_{1}$ functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). …##### 16: Bibliography M

##### 17: 32.10 Special Function Solutions

###### §32.10(iv) Fourth Painlevé Equation

… ►###### §32.10(vi) Sixth Painlevé Equation

… ►where the fundamental periods $2{\varphi}_{1}$ and $2{\varphi}_{2}$ are linearly independent functions satisfying the hypergeometric equation … ►##### 18: 19.16 Definitions

###### §19.16(ii) ${R}_{-a}(\mathbf{b};\mathbf{z})$

►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 $R$-function as a Dirichlet average, see Carlson (1977b). ►###### §19.16(iii) Various Cases of ${R}_{-a}(\mathbf{b};\mathbf{z})$

…##### 19: 2.6 Distributional Methods

*tempered distribution*(i. … ►Corresponding results for the

*generalized Stieltjes transform*… ►These equations again hold only in the sense of distributions. … ►For rigorous derivations of these results and also order estimates for ${\delta}_{n}(x)$, see Wong (1979) and Wong (1989, Chapter 6).

##### 20: Errata

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 $n$-argument Wronskian is given by $\mathcal{W}\left\{{w}_{1}(z),\mathrm{\dots},{w}_{n}(z)\right\}=det\left[{w}_{k}^{(j-1)}(z)\right]$, where $1\le j,k\le n$. Immediately below Equation (1.13.4), a sentence was added giving the definition of the $n$-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 $n$th-order differential equations. A reference to Ince (1926, §5.2) was added.

These equations have been generalized to include the additional cases of $\partial {J}_{-\nu}\left(z\right)/\partial \nu $, $\partial {I}_{-\nu}\left(z\right)/\partial \nu $, respectively.

There were clarifications made in the conditions on the parameter $a$ in $U(a,b,z)$ of those equations.

The Wronskian was generalized to include both associated Legendre and Ferrers functions.

has been generalized to cover an additional case.