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Author: Seppo Nurmi, Järfälla, Sweden.
Copyright © Seppo Nurmi, 1998, All Rights Reserved.
This work was originally written 29 June 1986.
Reworked on a MathCad worksheet December 1996.
Reworked as a HTML-document for Internet March 1998.

Ultraspherical Harmonics

1. Ultraspherical Coordinates

Let us consider spherical look-a-like orthogonal coordinates in an arbitrary number n of dimensions. Transformation into Cartesian coordinates can be expressed generally as follows:

(1.1)

where

(1.2)

Now, denoting formally so we have , we can write generally:

(1.3)

Define a set of generalised coordinates : (we give r the highest index for convenience)

and

for

(1.4)

We get the scalar-coefficients from the formula:

(1.5)

Observe that for all of the coordinates here is true:

(1.6)

Then for the n:th scalar coefficient

(1.7)

Further we get, if we observe that is independent of whenever , in general form

(1.8)

Elimination, using , starting from the two first rows, and working downwards

(1.9)

yields

(1.10)

So we conclude:

...

...

(1.11)

In a compact recursive notation we have for the scalar coefficients

(1.12)

starting from

(1.13)

Now denote

(1.14)

(1.15)

(1.16)

Then we get

(1.17)

and

(1.18)

Results, to be used in next chapter:

(1.19)

(1.20)

(1.21)

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