Editing Appendix/SpecialFunctions (section)

====Example Recurrence Relations====

The above [[#Analytic_Expressions_.26_Plots|''Toroidal Function Evaluations'']] table provides analytic expressions for the pair of foundation functions, <math>P^0_{-\frac{1}{2}}(z)</math> and <math>P^0_{+\frac{1}{2}}(z)</math>, and the associated pair of foundation functions, <math>Q^0_{-\frac{1}{2}}(z)</math> and <math>Q^0_{+\frac{1}{2}}(z)</math>.  From either pair of foundation functions, expressions for all other zero-order, half-integer degree toroidal functions can be obtained using a relatively simple recurrence relation drawn from the "Key Equation,"

{{ Math/EQ_Toroidal04 }}

Specifically, letting <math>\mu \rightarrow 0</math> and <math>\nu \rightarrow (m - \tfrac{1}{2})</math>, for all <math>~m \ge 2</math>, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P0_{m-\frac{1}{2}}(z)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4 \biggl[ \frac{m-1}{2m-1} \biggr] z P^0_{m-\frac{3}{2}}(z) - \biggl[ \frac{2m-3}{2m-1}\biggr]P^0_{m-\frac{5}{2}}(z) \, ;</math> &nbsp; &nbsp; and,
  </td>
</tr>

<tr>
  <td align="right">
<math>Q^0_{m-\frac{1}{2}}(z)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4 \biggl[ \frac{m-1}{2m-1} \biggr] z Q^0_{m-\frac{3}{2}}(z) - \biggl[ \frac{2m-3}{2m-1}\biggr]Q^0_{m-\frac{5}{2}}(z) \, .</math>
  </td>
</tr>
</table>

As examples, these two relations have been used to generate columns of numbers in the [[#Comparison_with_Table_IX_from_MF53|comparison table shown below]] for, respectively, the toroidal functions, <math>P^0_{+\frac{3}{2}}(z)</math> and <math>Q^0_{+\frac{3}{2}}(z)</math>. For order-1 and order-2 toroidal functions, the above table provides analytic expressions only for (the functions of the lowest half-integer degree) <math>Q^1_{-\frac{1}{2}}(z)</math> and  <math>Q^2_{-\frac{1}{2}}(z)</math>.  But, as we have detailed in an [[Appendix/Mathematics/ToroidalSynopsis01#Evaluating_Q2.CE.BD|accompanying discussion]], additional order-1 and order-2 expressions can be straightforwardly derived by drawing upon another key recurrence relation, namely,

{{ Math/EQ_Toroidal07 }}

Specifically, after adopting the association, <math>\nu \rightarrow (n - \tfrac{1}{2})</math>, we have, when <math>\mu = 0</math>,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>Q_{n - \frac{1}{2}}^{1}(z)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(n-\tfrac{1}{2}) (z^2-1)^{-\frac{1}{2}} [z Q_{n - \frac{1}{2}}(z) - Q_{n - \frac{3}{2}}(z)]
</math>
  </td>
  <td allign="center">&nbsp; &nbsp; &hellip; &nbsp; &nbsp;</td>
  <td align="left">
for <math>n \ge 1 \, ,</math>
  </td>
</tr>
</table>

and, when <math>~\mu = 1</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>Q_{n - \frac{1}{2}}^{2}(z)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(z^2-1)^{-\frac{1}{2}} \{ (n-\tfrac{3}{2}) z Q^1_{n - \frac{1}{2}}(z) - (n+\tfrac{1}{2})Q^1_{n - \frac{3}{2}}(z)\}
</math>
  </td>
  <td allign="center">&nbsp; &nbsp; &hellip; &nbsp; &nbsp;</td>
  <td align="left">
for <math>n \ge 1 \, .</math>
  </td>
</tr>
</table>
As an example, the first of these two relations has been used to generate a column of numbers in the [[#Comparison_with_Table_IX_from_MF53|comparison table shown below]] for the toroidal function, <math>Q^1_{+\frac{1}{2}}(z)</math>.