DEtools/hypergeometricsols - Maple Help

DEtools

 hypergeometricsols
 find closed form solutions of hypergeometric type for linear ODEs

 Calling Sequence hypergeometricsols(lode, v)

Parameters

 lode - linear ODE in diff form or operator form v - dependent variable of lode, or a list with the names of the dependent and independent variables

Description

 • hypergeometricsols contains numerous algorithms to find closed form solutions of Linear Ordinary Differential Equations with polynomial coefficients. These algorithms find, if they exist, solutions that can be written in terms of hypergeometric pFq functions. At the moment, hypergeometricsols only contains algorithms for equations of order two. There are three types of pFq solutions for order two, namely 0F1, 1F1, and 2F1.
 • Solutions are expressed in terms of hypergeometric pFq functions, exp, integrals, and algebraic functions. Such expressions are useful because pFq functions are rigid, which means that their global behavior is fully determined exactly by local data (for example, a 2F1 type solution gives the exact global monodromy).
 • hypergeometricsols contains complete algorithms, from references (1) and (2) below, that find any solution that can be expressed in terms of 0F1 or 1F1 type functions (written in terms of Airy/Bessel resp. Kummer/Whittaker functions). It computes Liouvillian solutions with the algorithm from (3), and 2F1 type solutions with the algorithms from (4),(5),(6),(7).
 • hypergeometricsols may return more than two 2F1 type solutions (the first two will be independent). The reason for doing this is because different expressions may converge in different parts of the complex plane. There are many relations between 2F1 functions, so there can be many 2F1 type solutions; hypergeometricsols attempts to make a reasonable selection.

Userinfo and singularities

 • Setting infolevel[hypergeometricsols] to a positive integer will cause hypergeometricsols to give information during the computation, such as the number of singularities of each type. Two equations L1 and L2 are considered to be exp-equivalent if there exists a rational function r such that {all solutions of L1} = exp(int(r,x))  {all solutions of L2}. hypergeometricsols will only count a point x=p as a singular point if x=p stays singular under exp-equivalence.
 • Such p is considered an apparent singularity if all solutions become analytic at x = p after some exp-equivalence. If x=p becomes regular singular under exp-equivalence, but not non-singular or apparent, then x=p is counted as a regular singularity, and in the remaining case, it is an irregular singularity. With this way of counting, if there is any irregular singularity, then no 2F1-type solutions exist. Similarly, if there is no irregular singularity, then no 0F1/1F1 type solution exists. An equation with 3 regular singularities and 0 other singularities will be referred to as a Gauss Hypergeometric Equation (even if the singularities are not located in their standard locations {0, 1, infinity}). Likewise, an equation with 4 regular and 0 other singularities is considered a Heun equation. hypergeometricsols has tables from references (4) and (5) to quickly handle common equations with up to 5 regular singularities (plus any number of apparent singularities). The remaining 2F1 cases are handled by the algorithms from (6) and (7).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$

Maple can now solve any second order equation that can be solved in terms of 0F1 or 1F1 functions. As an example, the following equation, has a solution that is a linear combination of two Bessel type functions (0F1) with rational coefficients

 > $\mathrm{_Envdiffopdomain}≔\left[\mathrm{Dx},x\right]:$
 > $L≔\left(2x-9\right)\left(2x-15\right){\mathrm{Dx}}^{2}+\left(-8x+48\right)\mathrm{Dx}-\frac{4{x}^{6}-96{x}^{5}+931{x}^{4}-4452{x}^{3}+10539{x}^{2}-12474x+9720}{4{x}^{2}{\left(x-3\right)}^{2}}$
 ${L}{≔}\left({2}{}{x}{-}{9}\right){}\left({2}{}{x}{-}{15}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({-}{8}{}{x}{+}{48}\right){}{\mathrm{Dx}}{-}\frac{{4}{}{{x}}^{{6}}{-}{96}{}{{x}}^{{5}}{+}{931}{}{{x}}^{{4}}{-}{4452}{}{{x}}^{{3}}{+}{10539}{}{{x}}^{{2}}{-}{12474}{}{x}{+}{9720}}{{4}{}{{x}}^{{2}}{}{\left({x}{-}{3}\right)}^{{2}}}$ (1)
 > $\mathrm{hypergeometricsols}\left(L\right)$
 $\left[\frac{{x}{}\left(\left({4}{}{{x}}^{{3}}{-}{36}{}{{x}}^{{2}}{+}{141}{}{x}{-}{225}\right){}{\mathrm{BesselI}}{}\left({1}{,}\frac{{x}}{{2}}\right){+}\left({-}{4}{}{{x}}^{{3}}{+}{24}{}{{x}}^{{2}}{-}{63}{}{x}{+}{27}\right){}{\mathrm{BesselI}}{}\left({2}{,}\frac{{x}}{{2}}\right)\right)}{{1}{-}\frac{{x}}{{3}}}{,}\frac{{x}{}\left(\left({-}{4}{}{{x}}^{{3}}{+}{36}{}{{x}}^{{2}}{-}{141}{}{x}{+}{225}\right){}{\mathrm{BesselK}}{}\left({1}{,}\frac{{x}}{{2}}\right){+}\left({-}{4}{}{{x}}^{{3}}{+}{24}{}{{x}}^{{2}}{-}{63}{}{x}{+}{27}\right){}{\mathrm{BesselK}}{}\left({2}{,}\frac{{x}}{{2}}\right)\right)}{{1}{-}\frac{{x}}{{3}}}\right]$ (2)

The previous example was entered as a differential operator. hypergeometricsols also accepts a second order differential equation as input. This example has a 1F1 type solution:

 > $L≔4x\left(4{x}^{3}-4{x}^{2}+8x-1\right){\mathrm{Dx}}^{2}+\left(-16{x}^{4}-32{x}^{2}+36x-8\right)\mathrm{Dx}+12{x}^{3}+4{x}^{2}+12x-27$
 ${L}{≔}{4}{}{x}{}\left({4}{}{{x}}^{{3}}{-}{4}{}{{x}}^{{2}}{+}{8}{}{x}{-}{1}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({-}{16}{}{{x}}^{{4}}{-}{32}{}{{x}}^{{2}}{+}{36}{}{x}{-}{8}\right){}{\mathrm{Dx}}{+}{12}{}{{x}}^{{3}}{+}{4}{}{{x}}^{{2}}{+}{12}{}{x}{-}{27}$ (3)
 > $\mathrm{ode}≔\mathrm{diffop2de}\left(L,y\left(x\right)\right)$
 ${\mathrm{ode}}{≔}\left({12}{}{{x}}^{{3}}{+}{4}{}{{x}}^{{2}}{+}{12}{}{x}{-}{27}\right){}{y}{}\left({x}\right){+}\left({-}{16}{}{{x}}^{{4}}{-}{32}{}{{x}}^{{2}}{+}{36}{}{x}{-}{8}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{4}{}{x}{}\left({4}{}{{x}}^{{3}}{-}{4}{}{{x}}^{{2}}{+}{8}{}{x}{-}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (4)
 > $\mathrm{hypergeometricsols}\left(\mathrm{ode}\right)$
 $\left[\left({-}{20}{}{x}{+}{5}\right){}{\mathrm{KummerM}}{}\left(\frac{{9}}{{4}}{,}{3}{,}{x}\right){+}\left({12}{}{{x}}^{{2}}{-}{12}{}{x}{+}{3}\right){}{\mathrm{KummerM}}{}\left(\frac{{5}}{{4}}{,}{3}{,}{x}\right){,}\left({20}{}{x}{-}{5}\right){}{\mathrm{KummerU}}{}\left(\frac{{9}}{{4}}{,}{3}{,}{x}\right){+}\left({16}{}{{x}}^{{2}}{-}{16}{}{x}{+}{4}\right){}{\mathrm{KummerU}}{}\left(\frac{{5}}{{4}}{,}{3}{,}{x}\right)\right]$ (5)

All the solutions computable with hypergeometricsols are also computable using the general ODE solver, dsolve

 > $\mathrm{dsolve}\left(\mathrm{ode}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left({12}{}{\left({x}{-}\frac{{1}}{{2}}\right)}^{{2}}{}{\mathrm{KummerM}}{}\left(\frac{{5}}{{4}}{,}{3}{,}{x}\right){+}\left({-}{20}{}{x}{+}{5}\right){}{\mathrm{KummerM}}{}\left(\frac{{9}}{{4}}{,}{3}{,}{x}\right)\right){+}{\mathrm{_C2}}{}\left({16}{}{\left({x}{-}\frac{{1}}{{2}}\right)}^{{2}}{}{\mathrm{KummerU}}{}\left(\frac{{5}}{{4}}{,}{3}{,}{x}\right){+}\left({20}{}{x}{-}{5}\right){}{\mathrm{KummerU}}{}\left(\frac{{9}}{{4}}{,}{3}{,}{x}\right)\right)$ (6)

The following equations can be solved in in terms of Heun functions, and also in terms of pFq functions.

 > $L≔-4{x}^{4}+{\mathrm{Dx}}^{2}+4{x}^{2}-10x+2$
 ${L}{≔}{-}{4}{}{{x}}^{{4}}{+}{{\mathrm{Dx}}}^{{2}}{+}{4}{}{{x}}^{{2}}{-}{10}{}{x}{+}{2}$ (7)
 > $\mathrm{hypergeometricsols}\left(L\right)$
 $\left[\left({2}{}{x}{+}{1}\right){}{\mathrm{AiryAi}}{}\left({1}{,}{{x}}^{{2}}{-}{1}\right){+}\left({2}{}{{x}}^{{2}}{+}{x}{-}{1}\right){}{\mathrm{AiryAi}}{}\left({{x}}^{{2}}{-}{1}\right){,}\left({2}{}{x}{+}{1}\right){}{\mathrm{AiryBi}}{}\left({1}{,}{{x}}^{{2}}{-}{1}\right){+}\left({2}{}{{x}}^{{2}}{+}{x}{-}{1}\right){}{\mathrm{AiryBi}}{}\left({{x}}^{{2}}{-}{1}\right)\right]$ (8)
 > $\mathrm{ode}≔\mathrm{diffop2de}\left(L,y\left(x\right)\right)$
 ${\mathrm{ode}}{≔}\left({-}{4}{}{{x}}^{{4}}{+}{4}{}{{x}}^{{2}}{-}{10}{}{x}{+}{2}\right){}{y}{}\left({x}\right){+}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)$ (9)
 > $\mathrm{Heunsols}\left(\mathrm{ode}\right)$
 $\left[{{ⅇ}}^{{-}\frac{{x}{}\left({2}{}{{x}}^{{2}}{-}{3}\right)}{{3}}}{}{\mathrm{HeunT}}{}\left(\frac{{3}{}{{6}}^{{2}}{{3}}}}{{4}}{,}{-}\frac{{15}}{{2}}{,}{-}{{6}}^{{1}}{{3}}}{,}\frac{{x}{}{{6}}^{{2}}{{3}}}}{{3}}\right){,}{{ⅇ}}^{\frac{{x}{}\left({2}{}{{x}}^{{2}}{-}{3}\right)}{{3}}}{}{\mathrm{HeunT}}{}\left(\frac{{3}{}{{6}}^{{2}}{{3}}}}{{4}}{,}\frac{{15}}{{2}}{,}{-}{{6}}^{{1}}{{3}}}{,}{-}\frac{{x}{}{{6}}^{{2}}{{3}}}}{{3}}\right)\right]$ (10)

By default, in this case, dsolve uses Heun functions

 > $\mathrm{dsolve}\left(\mathrm{ode}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{-}\frac{{2}}{{3}}{}{{x}}^{{3}}{+}{x}}{}{\mathrm{HeunT}}{}\left(\frac{{3}{}{{6}}^{{2}}{{3}}}}{{4}}{,}{-}\frac{{15}}{{2}}{,}{-}{{6}}^{{1}}{{3}}}{,}\frac{{x}{}{{6}}^{{2}}{{3}}}}{{3}}\right){+}{\mathrm{_C2}}{}{{ⅇ}}^{\frac{{2}}{{3}}{}{{x}}^{{3}}{-}{x}}{}{\mathrm{HeunT}}{}\left(\frac{{3}{}{{6}}^{{2}}{{3}}}}{{4}}{,}\frac{{15}}{{2}}{,}{-}{{6}}^{{1}}{{3}}}{,}{-}\frac{{x}{}{{6}}^{{2}}{{3}}}}{{3}}\right)$ (11)

You can use the method specification of dsolve to indicate the preferred functions

 > $\mathrm{dsolve}\left(\mathrm{ode},\left[\mathrm{Heun}\right]\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{-}\frac{{2}}{{3}}{}{{x}}^{{3}}{+}{x}}{}{\mathrm{HeunT}}{}\left(\frac{{3}{}{{6}}^{{2}}{{3}}}}{{4}}{,}{-}\frac{{15}}{{2}}{,}{-}{{6}}^{{1}}{{3}}}{,}\frac{{x}{}{{6}}^{{2}}{{3}}}}{{3}}\right){+}{\mathrm{_C2}}{}{{ⅇ}}^{\frac{{2}}{{3}}{}{{x}}^{{3}}{-}{x}}{}{\mathrm{HeunT}}{}\left(\frac{{3}{}{{6}}^{{2}}{{3}}}}{{4}}{,}\frac{{15}}{{2}}{,}{-}{{6}}^{{1}}{{3}}}{,}{-}\frac{{x}{}{{6}}^{{2}}{{3}}}}{{3}}\right)$ (12)
 > $\mathrm{dsolve}\left(\mathrm{ode},\left[\mathrm{hypergeometricsols}\right]\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left(\left({2}{}{x}{+}{1}\right){}{\mathrm{AiryAi}}{}\left({1}{,}{{x}}^{{2}}{-}{1}\right){+}\left({2}{}{{x}}^{{2}}{+}{x}{-}{1}\right){}{\mathrm{AiryAi}}{}\left({{x}}^{{2}}{-}{1}\right)\right){+}{\mathrm{_C2}}{}\left(\left({2}{}{x}{+}{1}\right){}{\mathrm{AiryBi}}{}\left({1}{,}{{x}}^{{2}}{-}{1}\right){+}\left({2}{}{{x}}^{{2}}{+}{x}{-}{1}\right){}{\mathrm{AiryBi}}{}\left({{x}}^{{2}}{-}{1}\right)\right)$ (13)

Note that references (1)+(2) give complete algorithms that are fully implemented in hypergeometricsols, so if hypergeometricsols fails to find 0F1 or 1F1 type solutions then such solutions do not exist.

 > $L≔3x{\mathrm{Dx}}^{2}+\left({x}^{2}+3x+4\right)\mathrm{Dx}-\frac{x+1}{3}$
 ${L}{≔}{3}{}{x}{}{{\mathrm{Dx}}}^{{2}}{+}\left({{x}}^{{2}}{+}{3}{}{x}{+}{4}\right){}{\mathrm{Dx}}{-}\frac{{x}}{{3}}{-}\frac{{1}}{{3}}$ (14)
 > $\mathrm{hypergeometricsols}\left(L\right)$
 $\left[\frac{{{ⅇ}}^{{-}\frac{{1}}{{12}}{}{{x}}^{{2}}{-}\frac{{1}}{{2}}{}{x}}{}\left(\left({x}{}\sqrt{\left({4}{+}{x}\right){}{x}}{+}{4}{}\sqrt{\left({4}{+}{x}\right){}{x}}\right){}{\mathrm{BesselI}}{}\left(\frac{{4}}{{3}}{,}\left(\frac{{1}}{{3}}{+}\frac{{x}}{{12}}\right){}\sqrt{\left({4}{+}{x}\right){}{x}}\right){+}\left({{x}}^{{2}}{+}{6}{}{x}{+}{16}\right){}{\mathrm{BesselI}}{}\left(\frac{{1}}{{3}}{,}\left(\frac{{1}}{{3}}{+}\frac{{x}}{{12}}\right){}\sqrt{\left({4}{+}{x}\right){}{x}}\right)\right)}{{{x}}^{{1}}{{6}}}{}\sqrt{{1}{+}\frac{{x}}{{4}}}}{,}\frac{{{ⅇ}}^{{-}\frac{{1}}{{12}}{}{{x}}^{{2}}{-}\frac{{1}}{{2}}{}{x}}{}\left(\left({-}{x}{}\sqrt{\left({4}{+}{x}\right){}{x}}{-}{4}{}\sqrt{\left({4}{+}{x}\right){}{x}}\right){}{\mathrm{BesselK}}{}\left(\frac{{4}}{{3}}{,}\left(\frac{{1}}{{3}}{+}\frac{{x}}{{12}}\right){}\sqrt{\left({4}{+}{x}\right){}{x}}\right){+}\left({{x}}^{{2}}{+}{6}{}{x}{+}{16}\right){}{\mathrm{BesselK}}{}\left(\frac{{1}}{{3}}{,}\left(\frac{{1}}{{3}}{+}\frac{{x}}{{12}}\right){}\sqrt{\left({4}{+}{x}\right){}{x}}\right)\right)}{{{x}}^{{1}}{{6}}}{}\sqrt{{1}{+}\frac{{x}}{{4}}}}\right]$ (15)

The above example shows it is possible for a small equation to have a complicated pFq type solution. An non-trivial examples with 2F1 solutions

 > $L≔144\left(s{x}^{3}-s{x}^{2}+1\right)\left(x-1\right)\left(3x-2\right){\mathrm{Dx}}^{2}+\left(-1296as{x}^{4}+3024as{x}^{3}+1080s{x}^{4}-2304as{x}^{2}-2520s{x}^{3}+576asx+2016s{x}^{2}-576sx-216x+288\right)\mathrm{Dx}+s{\left(3x-2\right)}^{3}\left(-1+6a\right)\left(6a-5\right)$
 ${L}{≔}{144}{}\left({s}{}{{x}}^{{3}}{-}{s}{}{{x}}^{{2}}{+}{1}\right){}\left({x}{-}{1}\right){}\left({3}{}{x}{-}{2}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({-}{1296}{}{a}{}{s}{}{{x}}^{{4}}{+}{3024}{}{a}{}{s}{}{{x}}^{{3}}{+}{1080}{}{s}{}{{x}}^{{4}}{-}{2304}{}{a}{}{s}{}{{x}}^{{2}}{-}{2520}{}{s}{}{{x}}^{{3}}{+}{576}{}{a}{}{s}{}{x}{+}{2016}{}{s}{}{{x}}^{{2}}{-}{576}{}{s}{}{x}{-}{216}{}{x}{+}{288}\right){}{\mathrm{Dx}}{+}{s}{}{\left({3}{}{x}{-}{2}\right)}^{{3}}{}\left({-}{1}{+}{6}{}{a}\right){}\left({6}{}{a}{-}{5}\right)$ (16)
 > $\mathrm{hypergeometricsols}\left(L\right)$
 $\left[{\mathrm{hypergeom}}{}\left(\left[{-}\frac{{a}}{{2}}{+}\frac{{5}}{{12}}{,}\frac{{1}}{{12}}{-}\frac{{a}}{{2}}\right]{,}\left[\frac{{1}}{{2}}\right]{,}{-}{s}{}\left({x}{-}{1}\right){}{{x}}^{{2}}\right){,}{x}{}\sqrt{{1}{-}{x}}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{7}}{{12}}{-}\frac{{a}}{{2}}{,}\frac{{11}}{{12}}{-}\frac{{a}}{{2}}\right]{,}\left[\frac{{3}}{{2}}\right]{,}{-}{s}{}\left({x}{-}{1}\right){}{{x}}^{{2}}\right){,}{\mathrm{hypergeom}}{}\left(\left[{-}\frac{{a}}{{2}}{+}\frac{{5}}{{12}}{,}\frac{{1}}{{12}}{-}\frac{{a}}{{2}}\right]{,}\left[{-}{a}{+}{1}\right]{,}{s}{}{{x}}^{{3}}{-}{s}{}{{x}}^{{2}}{+}{1}\right){,}{\left({s}{}{{x}}^{{3}}{-}{s}{}{{x}}^{{2}}{+}{1}\right)}^{{a}}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{1}}{{12}}{+}\frac{{a}}{{2}}{,}\frac{{5}}{{12}}{+}\frac{{a}}{{2}}\right]{,}\left[{1}{+}{a}\right]{,}{s}{}{{x}}^{{3}}{-}{s}{}{{x}}^{{2}}{+}{1}\right)\right]$ (17)

hypergeometricsols returned more than two solutions here because it is not always obvious which 2F1 type solution is best. However, the first two entries will be linearly independent. Here is an example of a Heun type equation that Maple can now also solve in terms of 2F1 solutions:

 > $L≔64{x}^{2}\left(27{x}^{2}+14x+3\right){\mathrm{Dx}}^{2}+48\left(5x+1\right)\left(9x+1\right)x\mathrm{Dx}-3\left({b}^{2}-9\right){\left(x+1\right)}^{2}$
 ${L}{≔}{64}{}{{x}}^{{2}}{}\left({27}{}{{x}}^{{2}}{+}{14}{}{x}{+}{3}\right){}{{\mathrm{Dx}}}^{{2}}{+}{48}{}\left({5}{}{x}{+}{1}\right){}\left({9}{}{x}{+}{1}\right){}{x}{}{\mathrm{Dx}}{-}{3}{}\left({{b}}^{{2}}{-}{9}\right){}{\left({x}{+}{1}\right)}^{{2}}$ (18)

Turn ON infolevel

 > $\mathrm{infolevel}\left[\mathrm{hypergeometricsols}\right]≔1$
 ${{\mathrm{infolevel}}}_{{\mathrm{hypergeometricsols}}}{≔}{1}$ (19)
 > $\mathrm{sols}≔\mathrm{hypergeometricsols}\left(L\right):$
 DEtools/hypergeometricsols:   "Number of [regular, irregular, apparant] singularities is [4, 0, 0] (Heun Equation)" DEtools/hypergeometricsols:   "Exponent difference at regular singularities:"   [infinity, 1/12*b], [0, 1/4*b], [RootOf(27*_Z^2+14*_Z+3), 1/2] DEtools/hypergeometricsols:   "Trying 2F1 algorithms for common cases." DEtools/hypergeometricsols:   "Found non-Louivillian 2F1 solutions"
 > $\mathrm{sols}\left[1..2\right]$
 $\left[{\left({1}{+}\frac{{14}}{{3}}{}{x}{+}{9}{}{{x}}^{{2}}\right)}^{\frac{{b}}{{24}}{-}\frac{{1}}{{8}}}{}{{x}}^{{-}\frac{{b}}{{8}}{+}\frac{{3}}{{8}}}{}{\left({1}{-}\frac{{x}}{{3}}\right)}^{\frac{{b}}{{12}}{-}\frac{{1}}{{4}}}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{1}}{{8}}{-}\frac{{b}}{{24}}{,}\frac{{5}}{{8}}{-}\frac{{b}}{{24}}\right]{,}\left[{1}{-}\frac{{b}}{{12}}\right]{,}{-}\frac{{256}{}{{x}}^{{3}}}{{\left({x}{-}{3}\right)}^{{2}}{}\left({27}{}{{x}}^{{2}}{+}{14}{}{x}{+}{3}\right)}\right){,}{\left({1}{+}\frac{{14}}{{3}}{}{x}{+}{9}{}{{x}}^{{2}}\right)}^{{-}\frac{{b}}{{24}}{-}\frac{{1}}{{8}}}{}{{x}}^{\frac{{b}}{{8}}{+}\frac{{3}}{{8}}}{}{\left({1}{-}\frac{{x}}{{3}}\right)}^{{-}\frac{{b}}{{12}}{-}\frac{{1}}{{4}}}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{1}}{{8}}{+}\frac{{b}}{{24}}{,}\frac{{5}}{{8}}{+}\frac{{b}}{{24}}\right]{,}\left[{1}{+}\frac{{b}}{{12}}\right]{,}{-}\frac{{256}{}{{x}}^{{3}}}{{\left({x}{-}{3}\right)}^{{2}}{}\left({27}{}{{x}}^{{2}}{+}{14}{}{x}{+}{3}\right)}\right)\right]$ (20)

References

 (1) M. van Hoeij and Q. Yuan. {em Finding all Bessel type solutions for Linear Differential Equations with Rational Function Coefficients}, ISSAC'2010 Proceedings.
 (2) R. Debeerst, M. van Hoeij, W. Koepf, Solving Differential Equations in Terms of Bessel Functions, ISSAC'2008 Proceedings.
 (3)    M. van Hoeij, J-A. Weil, Solving Second Order Linear Differential Equations with Klein's Theorem. ISSAC'2005 Proceedings.
 (4)    R. Vidunas, G. Filipuk: A Classification of Covering yielding Heun to Hypergeometric Reductions, Funkcialaj Ekvacioj (2013).
 (5)    M. van Hoeij, V. Kunwar, Classifying (near)-Belyi maps with Five Exceptional Points. Indagationes Mathematicae (2019).
 (6) V. Kunwar and M. van Hoeij, Second Order Differential Equations with Hypergeometric Solutions of Degree Three, ISSAC'2013 Proceedings.
 (7) E. Imamoglu and M. van Hoeij, Computing Hypergeometric Solutions of Second Order Linear Differential Equations using Quotients of Formal Solutions and Integral Bases,  J. of Symbolic Computation, (2017).

Compatibility

 • The DEtools[hypergeometricsols] command was introduced in Maple 2021.