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1: Bille C. Carlson
In his paper Lauricella’s hypergeometric function F D (1963), he defined the R -function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter. …Also, the homogeneity of the R -function has led to a new type of mean value for several variables, accompanied by various inequalities. …
2: 3.6 Linear Difference Equations
If d n = 0 , n , then the difference equation is homogeneous; otherwise it is inhomogeneous. …
§3.6(ii) Homogeneous Equations
Because the recessive solution of a homogeneous equation is the fastest growing solution in the backward direction, it occurred to J. … See also Gautschi (1967) and Gil et al. (2007a, Chapter 4) for the computation of recessive solutions via continued fractions. … It is applicable equally to the computation of the recessive solution of the homogeneous equation (3.6.3) or the computation of any solution w n of the inhomogeneous equation (3.6.1) for which the conditions of §3.6(iv) are satisfied. …
3: 9.16 Physical Applications
An example from quantum mechanics is given in Landau and Lifshitz (1965), in which the exact solution of the Schrödinger equation for the motion of a particle in a homogeneous external field is expressed in terms of Ai ( x ) . …
4: 15.11 Riemann’s Differential Equation
The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). … 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 …
5: 19.33 Triaxial Ellipsoids
Let a homogeneous magnetic ellipsoid with semiaxes a , b , c , volume V = 4 π a b c / 3 , and susceptibility χ be placed in a previously uniform magnetic field H parallel to the principal axis with semiaxis c . … The same result holds for a homogeneous dielectric ellipsoid in an electric field. …
6: 20.9 Relations to Other Functions
7: 1.13 Differential Equations
§1.13(iii) Inhomogeneous Equations
If w 0 ( z ) is any one solution, and w 1 ( z ) , w 2 ( z ) are a fundamental pair of solutions of the corresponding homogeneous equation (1.13.1), then every solution of (1.13.8) can be expressed as … For extensions of these results to linear homogeneous differential equations of arbitrary order see Spigler (1984). …
8: 3.7 Ordinary Differential Equations
If h = 0 the differential equation is homogeneous, otherwise it is inhomogeneous. … … The equations can then be solved by the method of §3.2(ii), if the differential equation is homogeneous, or by Olver’s algorithm (§3.6(v)). …
9: 12.17 Physical Applications
Problems on high-frequency scattering in homogeneous media by parabolic cylinders lead to asymptotic methods for integrals involving PCFs. …
10: 23.10 Addition Theorems and Other Identities
§23.10(iv) Homogeneity