Hypergeometric functions $\mathrm{pFq}\left(a_1\,\mathrm{...}\,a_p\,b_1\,\mathrm{...}\,b_q\,f\left(x\right)\right)$, or their generalization known as MeijerG functions, can be defined in different ways. One possible way is to define them as the solutions to some linear ODEs. In connection with that approach, the hyperode routine receives as input a hypergeometric function H as a parameter, where H is constructed using one of hypergeom or MeijerG, as either

H([...], [...], f(x)), where $f\left(x\right)$ is an arbitrary expression depending on x, or

the product of an arbitrary expression $g\left(x\right)$ times H([...], [...], f(x)).

Using this input, hyperode returns the general nth order ODE having as a solution that hypergeometric function H or that product of H times $g\left(x\right)$.

By identifying the general ODE underlying the definition of a hypergeometric function, this command can be of help when studying properties of these relevant functions and facilitate the understanding of related algorithms. In other cases, this ODE representation of pFq or MeijerG opens the way to computations (for example, differential elimination) that can only be performed with polynomial differential objects. See dpolyform, casesplit, DifferentialAlgebra, and rifsimp.

The ODE returned by hyperode, having a solution $y\left(x\right)=g\left(x\right)\mathrm{pFq}\left(a_1\,\mathrm{...}\,a_p\,b_i\,\mathrm{...}\,b_q\,f\left(x\right)\right)$ (where $\mathrm{pFq}$ is expressed using hypergeom) is built by using the formula

where D is the differential operator $\frac{d}{\mathrm{dx}}$, followed by a change of variables

and finally by the renaming of the variables according to

The output in this case is a linear ODE where the differential order is the largest of $p$ and $q+1$.

The ODE returned by hyperode, when the input is a MeijerG generalized hypergeometric function, is built using the same process, but the formula used in the first step is:

where in above we assume, without loss of generality, that $p$ is less than or equal to $q$, and $a_i$ and $b_i$ represent, respectively, the parameters entering the first and second lists of parameters in MeijerG. The output in this case is a $q$th order linear ODE.

When the differential order of the ODE to be returned is not indicated, the one implied by the given pFq or MeijerG function is used. When the differential order is indicated as an extra argument, and provided it is greater than the one implied by the hypergeometric function, the ODE is constructed in the same way as just outlined except that before proceeding, the lists $\[a,...\]$, $\[b,...\]$ are augmented by introducing into both of them the necessary number of additional parameters, all equal to zero.

Since the transformation shown above with arbitrary $f\left(x\right)$ and $g\left(x\right)$ represents the structure invariance group for the linear ODEs (that is, it is enough to map any linear ODE into any other one of the same differential order), then by keeping both $f\left(x\right)$ and $g\left(x\right)$ arbitrary, the ODE returned is "the most general linear ODE of a given order, written in such a way that its solution is expressed in terms of the product of a hypergeometric function (pFq or MeijerG) times an arbitrary function g(x)." Therefore, any linear ODE and its solution in terms of a hypergeometric function can be obtained from this general form by appropriately choosing f and g (see the last example).

When an optional list of additional ODE solutions $\[S_1\left(x\right),S_2\left(x\right),...,S_n\left(x\right)\]$ is given, the returned ODE is built by first using the formula explained above, and then applying to it operators of the form

for each $\mathrm{Sn}\left(x\right)$, where the functions $\mathrm{Fn}\left(x\right)$ are adjusted so that in addition to

for all $\mathrm{Sn}\left(x\right)$,

is also a solution of the returned ODE.

Note: This building process raises the differential order of the returned ODE by n.

with(DEtools):

PDEtools[declare](y(x), prime=x);

$y\left(x\right)\mathrm{will\; now\; be\; displayed\; as}y$

$\mathrm{derivatives\; with\; respect\; to}x\mathrm{of\; functions\; of\; one\; variable\; will\; now\; be\; displayed\; with\; \text{'}}$

hypergeom([a,b],[c],x);

$\mathrm{hypergeom}\left(\left[a\,b\right]\,\left[c\right]\,x\right)$

hyperode((2),y(x)) = 0;

$yab+\left(\left(a+b+1\right)x-c\right)\mathrm{y\text{'}}+\left(x^2-x\right)\mathrm{y\text{'}\text{'}}=0$

hypergeom([a],[],x);

$\mathrm{hypergeom}\left(\left[a\right]\,\left[\right]\,x\right)$

hyperode((4),y(x));

$ay+\left(x-1\right)\mathrm{y\text{'}}$

dsolve((5));

$y=\mathrm{c\_\_1}{\left(x-1\right)}^{-a}$

convert(hypergeom([a],[],x), Sum, x) assuming abs(x)<1;

$\sum _{\mathrm{\_k1}=0}^{\mathrm{\infty }}\frac{\mathrm{pochhammer}\left(a\&comma;\mathrm{\_k1}\right)x^{\mathrm{\_k1}}}{\mathrm{\_k1}!}$

value((7)) assuming abs(x)<1;

$\frac{1}{{\left(1-x\right)}^a}$

convert(hypergeom([a],[],x), StandardFunctions);

${\left(1-x\right)}^{-a}$

hypergeom([],[],x);

$\mathrm{hypergeom}\left(\left[\right]\&comma;\left[\right]\&comma;x\right)$

hyperode((10),y(x));

$y-\mathrm{y\text{'}}$

dsolve((11));

$y=\mathrm{c\_\_1}{\&ExponentialE;}^x$

hypergeom([],[1],x^2/4);

$\mathrm{hypergeom}\left(\left[\right]\&comma;\left[1\right]\&comma;\frac{x^2}{4}\right)$

hyperode((13),y(x));

$-x\mathrm{y\text{'}\text{'}}+yx-\mathrm{y\text{'}}$

dsolve((14));

$y=\mathrm{c\_\_1}\mathrm{BesselI}\left(0\&comma;x\right)+\mathrm{c\_\_2}\mathrm{BesselK}\left(0\&comma;x\right)$

convert((13),StandardFunctions);

$\mathrm{BesselI}\left(0\&comma;x\right)$

hypergeom([a,b, r],[c, s],x);

$\mathrm{hypergeom}\left(\left[a\&comma;b\&comma;r\right]\&comma;\left[c\&comma;s\right]\&comma;x\right)$

hyperode((17),y(x)) = 0;

odetest(y(x)=(17), (18));

$0$

hypergeom([1/3,2/7],[3/5], x);

$\mathrm{hypergeom}\left(\left[\frac{2}{7}\&comma;\frac{1}{3}\right]\&comma;\left[\frac{3}{5}\right]\&comma;x\right)$

hyperode((20),y(x), [exp(x)]) = 0;

odetest( y(x)=(20), (21));

$0$

odetest( y(x)=exp(x), (21));

$0$

MeijerG( [[a],[b]], [[c],[d]], x);

$\mathrm{MeijerG}\left(\left[\left[a\right]\&comma;\left[b\right]\right]\&comma;\left[\left[c\right]\&comma;\left[d\right]\right]\&comma;x\right)$

hyperode((24), y(x)) = 0;

hyperode( hypergeom( [a,b], [c],x), y(x) ) = 0;

$yab+\left(\left(a+b+1\right)x-c\right)\mathrm{y\text{'}}+\left(x^2-x\right)\mathrm{y\text{'}\text{'}}=0$

hypergeom([a,b],[c],f(x));

$\mathrm{hypergeom}\left(\left[a\&comma;b\right]\&comma;\left[c\right]\&comma;f\left(x\right)\right)$

(27) = convert( (27), MeijerG);

$\mathrm{hypergeom}\left(\left[a\&comma;b\right]\&comma;\left[c\right]\&comma;f\left(x\right)\right)=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{MeijerG}\left(\left[\left[1-a\&comma;-b+1\right]\&comma;\left[\right]\right]\&comma;\left[\left[0\right]\&comma;\left[1-c\right]\right]\&comma;-f\left(x\right)\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}$

PDEtools[declare](f(x),g(x));

$f\left(x\right)\mathrm{will\; now\; be\; displayed\; as}f$

$g\left(x\right)\mathrm{will\; now\; be\; displayed\; as}g$

f(x),g(x);

$f,g$

sol := y(x) = g(x) * hypergeom([a,b],[c],f(x));

$\mathrm{sol}≔y=g\mathrm{hypergeom}\left(\left[a\&comma;b\right]\&comma;\left[c\right]\&comma;f\right)$

hyperode(rhs(sol),y(x)) = 0;

$\left(\left({\mathrm{f\; \text{'}}}^6{g}^{4}b-f{g}^3\mathrm{g\text{'}}{\mathrm{f\; \text{'}}}^5\right)a-f{g}^3\mathrm{g\text{'}}{\mathrm{f\; \text{'}}}^{5}b+\mathrm{g\text{'}}{g}^3{\mathrm{f\; \text{'}}}^{5}c+f\left(fg\mathrm{g\text{'}}\mathrm{f\; \text{'}\text{'}}-fg\mathrm{f\; \text{'}}\mathrm{g\text{'}\text{'}}+2f{\mathrm{g\text{'}}}^2\mathrm{f\; \text{'}}-\mathrm{g\text{'}}g{\mathrm{f\; \text{'}}}^2-\mathrm{g\text{'}}\mathrm{f\; \text{'}\text{'}}g+g\mathrm{f\; \text{'}}\mathrm{g\text{'}\text{'}}-2{\mathrm{g\text{'}}}^2\mathrm{f\; \text{'}}\right)g^2{\mathrm{f\; \text{'}}}^3\right)y+\left(g^4{\mathrm{f\; \text{'}}}^{5}fa+g^4{\mathrm{f\; \text{'}}}^{5}fb-{\mathrm{f\; \text{'}}}^5{g}^{4}c-{\mathrm{f\; \text{'}}}^3{g}^{3}f\left(fg\mathrm{f\; \text{'}\text{'}}+2f\mathrm{g\text{'}}\mathrm{f\; \text{'}}-g{\mathrm{f\; \text{'}}}^2-g\mathrm{f\; \text{'}\text{'}}-2\mathrm{g\text{'}}\mathrm{f\; \text{'}}\right)\right)\mathrm{y\text{'}}+{\mathrm{f\; \text{'}}}^4{g}^{4}f\left(f-1\right)\mathrm{y\text{'}\text{'}}=0$

ode := collect(isolate( (32), diff(y(x),x,x)), diff, normal);

$\mathrm{ode}≔\mathrm{y\text{'}\text{'}}=\frac{\left(-\frac{y\mathrm{g\text{'}}}{g}+\mathrm{y\text{'}}\right)\mathrm{f\; \text{'}\text{'}}}{\mathrm{f\; \text{'}}}+\frac{y\mathrm{g\text{'}\text{'}}}{g}-\frac{bay{\mathrm{f\; \text{'}}}^2}{f\left(f-1\right)}+\left(\frac{\left(fa+fb+f-c\right)y\mathrm{g\text{'}}}{gf\left(f-1\right)}-\frac{\left(fa+fb+f-c\right)\mathrm{y\text{'}}}{f\left(f-1\right)}\right)\mathrm{f\; \text{'}}-\frac{2y{\mathrm{g\text{'}}}^2}{g^2}+\frac{2\mathrm{y\text{'}}\mathrm{g\text{'}}}{g}$

f=(x -> x+1/x), g=(x -> 1);

$f=\left(x\mapsto x+\frac{1}{x}\right),g=\left(x\mapsto 1\right)$

collect( eval(ode,{(34)}), diff, normal);

eval(sol, {(34)});

$y=\mathrm{hypergeom}\left(\left[a\&comma;b\right]\&comma;\left[c\right]\&comma;x+\frac{1}{x}\right)$

odetest( (36), (35) );

$0$

f=(x -> x^(2/3)+1), g=(x -> 1/(x^(2/3)+1));

$f=\left(x\mapsto x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+1\right),g=\left(x\mapsto \frac{1}{x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+1}\right)$

collect( eval(ode,{(38)}), diff, normal);

$\mathrm{y\text{'}\text{'}}=-\frac{\left(2xa+2xb+7x+2x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}a+2x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}b-2x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}c+3x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)\mathrm{y\text{'}}}{3x^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+1\right)}-\frac{4\left(x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}ab+x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}a+x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}b+x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+ab+a+b-c+1\right)y}{9{\left(x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+1\right)}^2{x}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

eval(sol, {(38)});

$y=\frac{\mathrm{hypergeom}\left(\left[a\&comma;b\right]\&comma;\left[c\right]\&comma;x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+1\right)}{x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+1}$

odetest((40), (39));

$0$

Marsden, J.E.; Sirovich, L.; and Antman, S.S. eds. Texts in Applied Mathematics. 56 vols. New York: Springer-Verlag, 1991. Vol. 8: Hypergeometric Functions and Their Applications.

Mathai, A.M. A Handbook of Generalized Special Functions for Statistical and Physical Sciences. Oxford: Clarendon Press, 1993.