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DEtools

 MeijerGsols
 solutions of a Meijer G type of linear ode

 Calling Sequence MeijerGsols(lode,v) MeijerGsols(coeff_list,x)

Parameters

 lode - homogeneous linear differential equation v - dependent variable of the lode coeff_list - list of coefficients of a linear ode x - independent variable of the lode

Description

 • The MeijerGsols routine returns a basis of the space of solutions of a linear differential equation of Meijer G type.
 • The classical notation used to represent the MeijerG function relates to the notation used in Maple by

${G}_{\mathrm{pq}}^{\mathrm{mn}}\left(z|\begin{array}{c}{a}_{1},...,{a}_{n},{a}_{n+1},...,{a}_{p}\\ {b}_{1},...,{b}_{m},{b}_{m+1},...,{b}_{q}\end{array}\right)$

$=\mathrm{MeijerG}\left(\left[\left[{a}_{1},\mathrm{...},{a}_{n}\right],\left[{a}_{n+1},\mathrm{...},{a}_{p}\right]\right],\left[\left[{b}_{1},\mathrm{...},{b}_{m}\right],\left[{b}_{m+1},\mathrm{...},{b}_{q}\right]\right],z\right)$

 Note: See Prudnikov, Brychkov, and Marichev.
 The MeijerG function satisfies the following qth-order linear differential equation

$\left({\left(-1\right)}^{p-m-n}x\left(\prod _{i=1}^{p}\left(x\mathrm{D}-{a}_{i}+1\right)\right)-\left(\prod _{i=1}^{q}\left(x\mathrm{D}-{b}_{i}\right)\right)\right)y\left(x\right)=0$

 where $'\mathrm{D}'=\frac{d}{\mathrm{dx}}$ and $p$ is less than or equal to q.
 • For example, MeijerG( [[a],[a[p]]], [[b],[b[q]]], x ) satisfies:
 > PDEtools[declare](y(x), prime=x);
 ${y}{}\left({x}\right){}{\mathrm{will now be displayed as}}{}{y}$
 ${\mathrm{derivatives with respect to}}{}{x}{}{\mathrm{of functions of one variable will now be displayed with \text{'}}}$ (1)
 > DEtools[hyperode]( MeijerG( [[a],[a[p]]], [[b],[b[q]]], x ), y(x) ) = 0;
 $\left(\left(\left({-}{{a}}_{{p}}{+}{1}\right){}{{a}}_{{1}}{+}{{a}}_{{p}}{-}{1}\right){}{x}{+}{{b}}_{{1}}{}{{b}}_{{q}}\right){}{y}{+}\left(\left({{a}}_{{1}}{+}{{a}}_{{p}}{-}{3}\right){}{{x}}^{{2}}{+}\left({-}{{b}}_{{1}}{-}{{b}}_{{q}}{+}{1}\right){}{x}\right){}{\mathrm{y\text{'}}}{+}\left({-}{{x}}^{{3}}{+}{{x}}^{{2}}\right){}{\mathrm{y\text{'}\text{'}}}{=}{0}$ (2)
 For information about making symbolic experiments with expressions that contain the MeijerG function of different arguments and the differential equation the expression satisfies, see dpolyform.
 • MeijerGsols accepts two calling sequences. The first argument of the first calling sequence is a linear differential equation in diff or $\mathrm{D}$ form, and the second argument is the function in the differential equation.
 • The first argument in the second calling sequence is the list of coefficients of a linear ode, and the second is the independent variable. This calling sequence can be convenient for programming with the MeijerGsols routine.
 • This function is part of the DEtools package, and so it can be used in the form MeijerGsols(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[MeijerGsols](..).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right),x,x\right)=\frac{4-2{x}^{2}}{{x}^{3}-x}\mathrm{diff}\left(y\left(x\right),x\right)-\frac{7}{4{x}^{4}-4{x}^{2}}y\left(x\right)$
 ${\mathrm{ode}}{≔}{\mathrm{y\text{'}\text{'}}}{=}\frac{\left({-}{2}{}{{x}}^{{2}}{+}{4}\right){}{\mathrm{y\text{'}}}}{{{x}}^{{3}}{-}{x}}{-}\frac{{7}{}{y}}{{4}{}{{x}}^{{4}}{-}{4}{}{{x}}^{{2}}}$ (3)
 > $\mathrm{MeijerGsols}\left(\mathrm{ode}\right)$
 $\left[{\mathrm{hypergeom}}{}\left(\left[{-}\frac{{7}}{{4}}{,}\frac{{1}}{{4}}\right]{,}\left[\frac{{1}}{{2}}\right]{,}\frac{{1}}{{{x}}^{{2}}}\right){,}\frac{{\mathrm{hypergeom}}{}\left(\left[{-}\frac{{5}}{{4}}{,}\frac{{3}}{{4}}\right]{,}\left[\frac{{3}}{{2}}\right]{,}\frac{{1}}{{{x}}^{{2}}}\right)}{{x}}\right]$ (4)

These routines for tackling MeijerG type linear ODEs can also be used directly from Maple's dsolve via

 > $\mathrm{dsolve}\left(\mathrm{ode},\left[\mathrm{MeijerG}\right]\right)$
 ${y}{=}{\mathrm{_C1}}{}{\mathrm{hypergeom}}{}\left(\left[{-}\frac{{7}}{{4}}{,}\frac{{1}}{{4}}\right]{,}\left[\frac{{1}}{{2}}\right]{,}\frac{{1}}{{{x}}^{{2}}}\right){+}\frac{{\mathrm{_C2}}{}{\mathrm{hypergeom}}{}\left(\left[{-}\frac{{5}}{{4}}{,}\frac{{3}}{{4}}\right]{,}\left[\frac{{3}}{{2}}\right]{,}\frac{{1}}{{{x}}^{{2}}}\right)}{{x}}$ (5)
 > $\mathrm{MeijerGsols}\left(\left[{x}^{2}-3,-x,{x}^{2}\right],x\right)$
 $\left[{x}{}{\mathrm{BesselJ}}{}\left({2}{,}{x}\right){,}{x}{}{\mathrm{BesselY}}{}\left({2}{,}{x}\right)\right]$ (6)
 > $A≔\mathrm{mult}\left(x\mathrm{DF}-\frac{1}{4},x\mathrm{DF}-1,x\mathrm{DF}-\frac{1}{3},\left[\mathrm{DF},x\right]\right)-{x}^{b}\mathrm{mult}\left(x\mathrm{DF}+\frac{2}{3},x\mathrm{DF}-\frac{1}{2},\left[\mathrm{DF},x\right]\right):$
 > $\mathrm{ode}≔\mathrm{diffop2de}\left(A,y\left(x\right),\left[\mathrm{DF},x\right]\right)$
 ${\mathrm{ode}}{≔}\left({-}\frac{{1}}{{12}}{+}\frac{{{x}}^{{b}}}{{3}}\right){}{y}{+}\left(\frac{{x}}{{12}}{-}\frac{{7}{}{{x}}^{{b}}{}{x}}{{6}}\right){}{\mathrm{y\text{'}}}{+}\left(\frac{{17}{}{{x}}^{{2}}}{{12}}{-}{{x}}^{{b}}{}{{x}}^{{2}}\right){}{\mathrm{y\text{'}\text{'}}}{+}{{x}}^{{3}}{}{\mathrm{y\text{'}\text{'}\text{'}}}$ (7)
 > $\mathrm{MeijerGsols}\left(\mathrm{ode}\right)$
 $\left[{x}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{5}}{{3}{}{b}}{,}\frac{{1}}{{2}{}{b}}\right]{,}\left[\frac{{4}{}{b}{+}{3}}{{4}{}{b}}{,}\frac{{3}{}{b}{+}{2}}{{3}{}{b}}\right]{,}\frac{{{x}}^{{b}}}{{b}}\right){,}{{x}}^{{1}}{{4}}}{}{\mathrm{hypergeom}}{}\left(\left[{-}\frac{{1}}{{4}{}{b}}{,}\frac{{11}}{{12}{}{b}}\right]{,}\left[\frac{{4}{}{b}{-}{3}}{{4}{}{b}}{,}\frac{{12}{}{b}{-}{1}}{{12}{}{b}}\right]{,}\frac{{{x}}^{{b}}}{{b}}\right){,}{{x}}^{{1}}{{3}}}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{1}}{{b}}{,}{-}\frac{{1}}{{6}{}{b}}\right]{,}\left[\frac{{3}{}{b}{-}{2}}{{3}{}{b}}{,}\frac{{12}{}{b}{+}{1}}{{12}{}{b}}\right]{,}\frac{{{x}}^{{b}}}{{b}}\right)\right]$ (8)

References

 Prudnikov, A. P.; Brychkov, Yu; and Marichev, O. Integrals and Series. Gordon and Breach Science, 1990. Vol. 3: More Special Functions.