Function Gegenbauer

Their solution, involving Gegenbauer functions, resulted in a wall correction factor for rigid spheres which is the same as the full expression corresponding to Eq. (32). 

These polynomial functions are closely related to the so-called Gegenbauer polynomials16 and satisfy the orthogonality condition... 

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... 

The Gegenbauer functions of the first kind are polynomials and can be represented in the form... 

The Gegenbauer functions of the second kind are given by the formulas... 

For n > 2, the Gegenbauer functions of the second kind are infinite at the points = 1, which correspond to 6 = 0 and 9 = it. Therefore, if there are no singularities in the physical statement of the problem, then all the constants with tildes in (2.1.5) must be zero. Moreover, for n = 0 and n = 1, the remaining constants, if nonzero, result in infinite values of the tangential velocity Vg on... 

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

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... 

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