The GenerateRecurrence command accepts one of the Appell functions and returns a procedure to compute the $n$th coefficient of a power series expansion of the given function F around the origin as a function of the $n-1$ previous coefficients. The recurrence is derived from the linear ODE underlying the function by assuming the existence of a power series solution for it. This means that only the first coefficients (initial conditions of the ODE) involve mathematical functions - all the other ones only involve arithmetic operations involving the value of those first coefficients, thus resulting in an efficient way of numerically computing the value of the function.

Within the context of that linear ODE, the value of the last argument of F is always ignored, taken to be a symbol, say $z$, and the linear ODE is constructed with $z$ as independent variable. To evaluate the function, the recurrence-procedure returned - say $u$- is used to generate the coefficient of a power series in $z$, as in $\sum _{n=0}^{\mathrm{\infty }}u\left(n\right)z^n$.

Depending on the value of the function's parameters (for example if they imply on a singular function, or the hypergeometric functions entering the initial conditions of the recurrence are divergent) it is not possible to construct a recurrence. In these cases GenerateRecurrence returns FAIL.

This approach, which within the Evalf package is referred as the series/recurrence approach, is one of the main methods for numerically evaluating mathematical functions and expressions where one or both of the following situations happen:

The underlying differential equation has singularities, as it is the case of most special functions of the mathematical language.

There is no closed form series expansion formula known (e.g. the Heun functions).

The recurrence returned by GenerateRecurrence can effectively compute the value of the given function provided that the following conditions are met:

The evaluation point $z$ where the function is to be numerically evaluated (the last argument of F) is not a singularity of the linear ODE underlying the given function F. To see the location of these singularities, use the Evalf:-Singularities command.

The evaluation point $z$ is within the radius of convergence of an expansion around the origin, i.e.: the distance from the origin to $z$ is smaller than the distance from the origin to the singularity closer to it.

The recurrence depends on $N$ initial conditions, $N\le d$ where $d$ is the order of the differential equation behind the function, that should be numerically computable by other means. These initial conditions are mainly the value of F and its first or also higher derivatives at the origin.

Generally speaking, when the recurrence can be used to evaluate a function at some (any) point $z_0$, by using a sequence of concatenated Taylor series expansions (that in turn requires knowing the recurrence formula for an expansion around an arbitrary $z_0$, that can also be derived from the underlying differential equation), the function can be computed over the whole complex plane.

When there are symbolic parameters (i.e.: not numbers) in the recurrence formula returned, it is frequently desired to simplify the result. For that purpose, use the option simplifier = ... where the right-hand side is the simplifier you want to be used. This simplification will be applied to the formula before constructing the procedure to be returned. Note however that this simplification is not applied to each coefficient each time the returned procedure is applied.

By default, the returned procedure is a function of the recursion index. If the given function F contains symbolic (i.e not a number) parameters, you may prefer to have the returned procedure being a function of these symbolic parameters as well. For that purpose, use the option parameters = [...], where the right-hand side is a list with the parameters to appear after the recursion index in the parameters of the returned procedure.

Initialization: Load the package and set the display of special functions in output to typeset mathematical notation (textbook notation):

$\mathrm{with}\left(\mathrm{MathematicalFunctions}:-\mathrm{Evalf}\right)\;\mathrm{Typesetting}:-\mathrm{EnableTypesetRule}\left(\mathrm{Typesetting}:-\mathrm{SpecialFunctionRules}\right)\:$

$\left\{\mathrm{Add}\,\mathrm{Evalb}\,\mathrm{Zoom}\,\mathrm{QuadrantNumbers}\,\mathrm{Singularities}\,\mathrm{GenerateRecurrence}\,\mathrm{PairwiseSummation}\right\}$

$\mathrm{F2}≔\mathrm{AppellF2}\left(1\,2\,3\,4\,5\,6\,\mathrm{z2}\right)$

$\mathrm{F2}≔F_2\left(1\,2\,3\,4\,5\,6\,\mathrm{z2}\right)$

$\mathrm{Singularities}\left(\mathrm{F2}\right)$

$\left[0\,1\,\mathrm{-5}\,\mathrm{\infty }+\mathrm{\infty }\mathrm{I}\right]$

$u≔\mathrm{GenerateRecurrence}\left(\mathrm{F2}\right)$

$u≔\mathbf{proc}\left(i\right)\mathbf{option}\mathrm{hfloat}\,\mathrm{cache}\;\mathrm{if}\left(i::\mathrm{nonnegint}\,4\/5\*\left(i\^2-4\right)\*\mathrm{thisproc}\left(i-1\right)\/\left(\left(i\+4\right)\*i\right)\+1\/5\*\left(i\^3-i\^2-10\*i-8\right)\*\mathrm{thisproc}\left(i-2\right)\/\left(i\*\left(i\+4\right)\*\left(i\+3\right)\right)\,\'\mathrm{procname}\left(\mathrm{args}\right)\'\right)\mathbf{end\; proc}$

$\mathrm{op}\left(4\,\mathrm{eval}\left(u\right)\right)$

$\mathrm{Cache}\left(512\,'\mathrm{permanent}'=\left[0={}_{2}F_1\left(1,2;4;6\right)\,1=\frac{3{}_{2}F_1\left(2,2;4;6\right)}{5}\,2=\frac{2{}_{2}F_1\left(2,3;4;6\right)}{5}\,3=\frac{2{}_{1}F_0\left(2;;6\right)}{7}\right]\right)$

$u\left(6\right)$

$\frac{188{}_{1}F_0\left(2;;6\right)}{9375}$

$\mathrm{op}\left(4\,\mathrm{eval}\left(u\right)\right)$

$\mathrm{Cache}\left(512\,'\mathrm{temporary}'=\left[4=\frac{3{}_{1}F_0\left(2;;6\right)}{35}\,5=\frac{29{}_{1}F_0\left(2;;6\right)}{750}\,6=\frac{188{}_{1}F_0\left(2;;6\right)}{9375}\right]\,'\mathrm{permanent}'=\left[0={}_{2}F_1\left(1,2;4;6\right)\,1=\frac{3{}_{2}F_1\left(2,2;4;6\right)}{5}\,2=\frac{2{}_{2}F_1\left(2,3;4;6\right)}{5}\,3=\frac{2{}_{1}F_0\left(2;;6\right)}{7}\right]\right)$

$u\left(n\right)$

$u\left(n\right)$

$\mathrm{evalf}\left(\right)$

$0.0008021333332$

$\mathrm{IF2}≔\mathrm{\%AppellF2}\left(1.0\,2.0\,3.0\,4.0\,5.0\,6.0\,\mathrm{z\_\_2}\right)$

$\mathrm{IF2}≔F_2\left(1.0\,2.0\,3.0\,4.0\,5.0\,6.0\,\mathrm{z\_\_2}\right)$

$u≔\mathrm{GenerateRecurrence}\left(\mathrm{IF2}\right)$

$u≔\mathbf{proc}\left(i\right)\mathbf{option}\mathrm{hfloat}\,\mathrm{cache}\;\mathrm{if}\left(i::\mathrm{nonnegint}\,\left(0.8\*i\^2-3.2\right)\*\mathrm{thisproc}\left(i-1\right)\/\left(\left(i\+4.\right)\*i\right)\+\left(0.2\*i\^3-0.2\*i\^2-2.\*i-1.6\right)\*\mathrm{thisproc}\left(i-2\right)\/\left(i\*\left(i\^2\+7.\*i\+12.\right)\right)\,\'\mathrm{procname}\left(\mathrm{args}\right)\'\right)\mathbf{end\; proc}$

$u\left(0\right)$

$\mathrm{-0.5568663767}-0.4363323130\mathrm{I}$

$u\left(6\right)$

$0.0008021333335+0.\mathrm{I}$

$\mathrm{op}\left(4\,\mathrm{eval}\left(u\right)\right)$

$\mathrm{Cache}\left(512\,'\mathrm{temporary}'=\left[4=0.003428571429+0.\mathrm{I}\,5=0.001546666667+0.\mathrm{I}\,6=0.0008021333335+0.\mathrm{I}\right]\,'\mathrm{permanent}'=\left[0=\mathrm{-0.5568663767}-0.4363323130\mathrm{I}\,1=\mathrm{-0.09270413916}+0.2094395102\mathrm{I}\,2=0.04454931012+0.03490658504\mathrm{I}\,3=0.01142857143\right]\right)$

The MathematicalFunctions[Evalf][GenerateRecurrence] command was introduced in Maple 2017.

For more information on Maple 2017 changes, see Updates in Maple 2017.