Gegenbauer polynomials

Equation (24) can be derived from the theory of hyperspherical harmonics and Gegenbauer polynomials but for readers unfamiliar with this theory, the expansion can be made plausible by substitution into the right-hand side of equation (23). With the help of the momentum-space orthonormality relations, (17), it can then be seen that right-hand side of (23) reduces to the left-hand side, which must be the case if the integral equation is to be satisfied. Let us now consider an electron moving in the attractive Coulomb potential of a collection of nuclei ... 

The radial behavior of the hydrogenic eigenfunctions in position and momentum space is exponential and Lorentzian , respectively, and their nodal structure depends on the associated Laguerre and Gegenbauer polynomials, respectively. 

Because the general solution (7 131) is expressed in terms of the modified Gegenbauer polynomials Q (r]), it is convenient to also express (7 146) in terms of these functions. Referring to definitions (7 132), we see that (7 146) can also be expressed in the form... 

Fock then expanded the kernel of this integral equation in terms of Gegenbauer polynomials and hyperspherical harmonics ... 

As a purely hydrodynamic problem, the velocity field due to a stagnant cap at the rear of a moving drop was solved exactly in terms of an infinite series of Gegenbauer polynomials with constants depending on the cap angle ( ). From this series, an analytical solution for the drag F(( > ) exerted on the drop can be obtained, from which the terminal velocity was computed once the external force on the drop is resolved,... 

An alternative to the Sack expansion is the defining equation of the Gegenbauer polynomials. 

Gegenbauer polynomials play a role in the theory of 3-dimensional spherical harmonics analogous to the role played by Legendre polynomials in the theory of 3-dimensional spherical harmonics and in fact, Legendre polynomials are a special case of Gegenbauer polynomials. For O = 3, we can recall the familiar expansion ... 

Like the Legendre polynomials, the Gegenbauer polynomials are found by expanding the generating function in a Taylor series, and collecting... 

Since the Gegenbauer polynomials are eigenfunctions of A, it must be possible to express them as linear combinations of hyperspherical harmonics belonging to the same eigenvalue ... 

In the theory of hyperspherical harmonics in a P-dimensional space, the Gegenbauer polynomials, C , with a = D/2 — 1, play a role analogous to that of the Legendre polynomials in the theory of three-dimensional spherical harmonics [4]. Since the Gegenbauer polynomials are the P-dimensional generalizations of Legendre polynomials, a possible choice of a P-dimensional multipole perturbation analogous to of equation (10), is... 

This simple proportionality for dipole perturbations does not carry over to other symmetries, i.e. the quadrupole term is a linear combination of two Gegenbauer polynomials. 

The resonance energy calculated up to Mth-order as a function of M is shown below. Note the oscillatory convergence towards the exact value of 2. The method used above to obtain 5," for n = 2,3,4 becomes extremely laborious for larger n. The results below were calculated by first showing that = 4PC where Q Gegenbauer polynomial of degree... 

Now in many problems, one wishes to express all the quantities in formula (13.4.5) in terms of polar coordinates u, rj, 6, dj, j, of the particles. In order to obtain a convenient formula for this case, one has to express Cn (cos a>t]) in terms of Gegenbauer pol3momials in the cosines of the polar angles. This is possible by usii the important addition theorem for Gegenbauer polynomials, which is given in the appendix of this chapter. Eq. (13.4.5) then becomes... 

The Gegenbauer polynomials arise quite naturally when one wants to expand any inverse power of the distance into a series in terms of angular variables. Such problems are very frequent when dealii with intermolecular forces, which are almost always expressed as such inverse powers of the intermolecular distances. One can say that Gegenbauer polynomials have the same importance in this field as Legendre polynomials in electrostatics. 

This formula is used to derive the two-centre expansion in Gegenbauer polynomials. 

In order to complete the proof of (13.4.10), one needs finally the expression of certain Gegenbauer polynomials in terms of... 

---