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.

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).

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

$\mathrm{\_Envdiffopdomain}≔\left[\mathrm{Dx}\,x\right]\:$

$L≔\left(2x-9\right)\left(2x-15\right){\mathrm{Dx}}^2+\left(-8x+48\right)\mathrm{Dx}-\frac{4x^6-96x^5+931x^4-4452x^3+10539x^2-12474x+9720}{4x^2{\left(x-3\right)}^2}$

$L≔\left(2x-9\right)\left(2x-15\right){\mathrm{Dx}}^2+\left(-8x+48\right)\mathrm{Dx}-\frac{4x^6-96x^5+931x^4-4452x^3+10539x^2-12474x+9720}{4x^2{\left(x-3\right)}^2}$

$\mathrm{hypergeometricsols}\left(L\right)$

$\left[\frac{x\left(\left(4x^3-36x^2+141x-225\right)\mathrm{BesselI}\left(1\,\frac{x}{2}\right)+\left(-4x^3+24x^2-63x+27\right)\mathrm{BesselI}\left(2\,\frac{x}{2}\right)\right)}{1-\frac{x}{3}}\,\frac{x\left(\left(-4x^3+36x^2-141x+225\right)\mathrm{BesselK}\left(1\,\frac{x}{2}\right)+\left(-4x^3+24x^2-63x+27\right)\mathrm{BesselK}\left(2\,\frac{x}{2}\right)\right)}{1-\frac{x}{3}}\right]$

$L≔4x\left(4x^3-4x^2+8x-1\right){\mathrm{Dx}}^2+\left(-16x^4-32x^2+36x-8\right)\mathrm{Dx}+12x^3+4x^2+12x-27$

$L≔4x\left(4x^3-4x^2+8x-1\right){\mathrm{Dx}}^2+\left(-16x^4-32x^2+36x-8\right)\mathrm{Dx}+12x^3+4x^2+12x-27$

$\mathrm{ode}≔\mathrm{diffop2de}\left(L\,y\left(x\right)\right)$

$\mathrm{ode}≔\left(12x^3+4x^2+12x-27\right)y\left(x\right)+\left(-16x^4-32x^2+36x-8\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+4x\left(4x^3-4x^2+8x-1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)$

$\mathrm{hypergeometricsols}\left(\mathrm{ode}\right)$

$\left[\left(-20x+5\right)\mathrm{KummerM}\left(\frac{9}{4}\,3\,x\right)+\left(12x^2-12x+3\right)\mathrm{KummerM}\left(\frac{5}{4}\,3\,x\right)\,\left(20x-5\right)\mathrm{KummerU}\left(\frac{9}{4}\,3\,x\right)+\left(16x^2-16x+4\right)\mathrm{KummerU}\left(\frac{5}{4}\,3\,x\right)\right]$

$\mathrm{dsolve}\left(\mathrm{ode}\right)$

$y\left(x\right)=\mathrm{c\_\_1}\left(12{\left(x-\frac{1}{2}\right)}^2\mathrm{KummerM}\left(\frac{5}{4}\,3\,x\right)+\left(-20x+5\right)\mathrm{KummerM}\left(\frac{9}{4}\,3\,x\right)\right)+\mathrm{c\_\_2}\left(16{\left(x-\frac{1}{2}\right)}^2\mathrm{KummerU}\left(\frac{5}{4}\,3\,x\right)+\left(20x-5\right)\mathrm{KummerU}\left(\frac{9}{4}\,3\,x\right)\right)$

$L≔-4x^4+{\mathrm{Dx}}^2+4x^2-10x+2$

$L≔-4x^4+{\mathrm{Dx}}^2+4x^2-10x+2$

$\mathrm{hypergeometricsols}\left(L\right)$

$\left[\left(2x+1\right)\mathrm{AiryAi}\left(1\,x^2-1\right)+\left(2x^2+x-1\right)\mathrm{AiryAi}\left(x^2-1\right)\,\left(2x+1\right)\mathrm{AiryBi}\left(1\,x^2-1\right)+\left(2x^2+x-1\right)\mathrm{AiryBi}\left(x^2-1\right)\right]$

$\mathrm{ode}≔\mathrm{diffop2de}\left(L\,y\left(x\right)\right)$

$\mathrm{ode}≔\left(-4x^4+4x^2-10x+2\right)y\left(x\right)+\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)$

$\mathrm{Heunsols}\left(\mathrm{ode}\right)$

$\left[{\ⅇ}^{-\frac{x\left(2x^2-3\right)}{3}}\mathrm{HeunT}\left(\frac{36^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{4}\,-\frac{15}{2}\,-6^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\,\frac{x{6}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3}\right)\,{\ⅇ}^{\frac{x\left(2x^2-3\right)}{3}}\mathrm{HeunT}\left(\frac{36^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{4}\,\frac{15}{2}\,-6^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\,-\frac{x{6}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3}\right)\right]$

$\mathrm{dsolve}\left(\mathrm{ode}\right)$

$y\left(x\right)=\mathrm{c\_\_1}{\ⅇ}^{-\frac{2}{3}{x}^3+x}\mathrm{HeunT}\left(\frac{36^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{4}\,-\frac{15}{2}\,-6^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\,\frac{x{6}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3}\right)+\mathrm{c\_\_2}{\ⅇ}^{\frac{2}{3}{x}^3-x}\mathrm{HeunT}\left(\frac{36^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{4}\,\frac{15}{2}\,-6^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\,-\frac{x{6}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3}\right)$

$\mathrm{dsolve}\left(\mathrm{ode}\,\left[\mathrm{Heun}\right]\right)$

$y\left(x\right)=\mathrm{c\_\_1}{\ⅇ}^{-\frac{2}{3}{x}^3+x}\mathrm{HeunT}\left(\frac{36^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{4}\,-\frac{15}{2}\,-6^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\,\frac{x{6}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3}\right)+\mathrm{c\_\_2}{\ⅇ}^{\frac{2}{3}{x}^3-x}\mathrm{HeunT}\left(\frac{36^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{4}\,\frac{15}{2}\,-6^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\,-\frac{x{6}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3}\right)$

$\mathrm{dsolve}\left(\mathrm{ode}\,\left[\mathrm{hypergeometricsols}\right]\right)$

$y\left(x\right)=\mathrm{c\_\_1}\left(\left(2x+1\right)\mathrm{AiryAi}\left(1\,x^2-1\right)+\left(2x^2+x-1\right)\mathrm{AiryAi}\left(x^2-1\right)\right)+\mathrm{c\_\_2}\left(\left(2x+1\right)\mathrm{AiryBi}\left(1\,x^2-1\right)+\left(2x^2+x-1\right)\mathrm{AiryBi}\left(x^2-1\right)\right)$

$L≔3x{\mathrm{Dx}}^2+\left(x^2+3x+4\right)\mathrm{Dx}-\frac{x+1}{3}$

$L≔3x{\mathrm{Dx}}^2+\left(x^2+3x+4\right)\mathrm{Dx}-\frac{x}{3}-\frac{1}{3}$

$\mathrm{hypergeometricsols}\left(L\right)$

$\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+6x+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^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\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+6x+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^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\sqrt{1+\frac{x}{4}}}\right]$

$L≔144\left(s{x}^3-s{x}^2+1\right)\left(x-1\right)\left(3x-2\right){\mathrm{Dx}}^2+\left(-1296as{x}^4+3024as{x}^3+1080s{x}^4-2304as{x}^2-2520s{x}^3+576asx+2016s{x}^2-576sx-216x+288\right)\mathrm{Dx}+s{\left(3x-2\right)}^3\left(-1+6a\right)\left(6a-5\right)$

$L≔144\left(s{x}^3-s{x}^2+1\right)\left(x-1\right)\left(3x-2\right){\mathrm{Dx}}^2+\left(-1296as{x}^4+3024as{x}^3+1080s{x}^4-2304as{x}^2-2520s{x}^3+576asx+2016s{x}^2-576sx-216x+288\right)\mathrm{Dx}+s{\left(3x-2\right)}^3\left(-1+6a\right)\left(6a-5\right)$

$\mathrm{hypergeometricsols}\left(L\right)$

$\left[\mathrm{hypergeom}\left(\left[-\frac{a}{2}+\frac{5}{12}\,\frac{1}{12}-\frac{a}{2}\right]\,\left[\frac{1}{2}\right]\,-s\left(x-1\right)x^2\right)\,x\sqrt{1-x}\mathrm{hypergeom}\left(\left[\frac{7}{12}-\frac{a}{2}\,\frac{11}{12}-\frac{a}{2}\right]\,\left[\frac{3}{2}\right]\,-s\left(x-1\right)x^2\right)\,\mathrm{hypergeom}\left(\left[-\frac{a}{2}+\frac{5}{12}\,\frac{1}{12}-\frac{a}{2}\right]\,\left[1-a\right]\,s{x}^3-s{x}^2+1\right)\,{\left(s{x}^3-s{x}^2+1\right)}^a\mathrm{hypergeom}\left(\left[\frac{1}{12}+\frac{a}{2}\,\frac{5}{12}+\frac{a}{2}\right]\,\left[1+a\right]\,s{x}^3-s{x}^2+1\right)\right]$

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

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

$\mathrm{infolevel}\left[\mathrm{hypergeometricsols}\right]≔1$

${\mathrm{infolevel}}_{\mathrm{hypergeometricsols}}≔1$

$\mathrm{sols}≔\mathrm{hypergeometricsols}\left(L\right)\:$

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"

$\mathrm{sols}\left[1..2\right]$

$\left[{\left(1-\frac{x}{3}\right)}^{\frac{b}{12}-\frac{1}{4}}{x}^{-\frac{b}{8}+\frac{3}{8}}{\left(1+\frac{14}{3}x+9x^2\right)}^{\frac{b}{24}-\frac{1}{8}}\mathrm{hypergeom}\left(\left[-\frac{b}{24}+\frac{1}{8}\,\frac{5}{8}-\frac{b}{24}\right]\,\left[1-\frac{b}{12}\right]\,-\frac{256x^3}{{\left(x-3\right)}^2\left(27x^2+14x+3\right)}\right)\,{\left(1-\frac{x}{3}\right)}^{-\frac{b}{12}-\frac{1}{4}}{x}^{\frac{b}{8}+\frac{3}{8}}{\left(1+\frac{14}{3}x+9x^2\right)}^{-\frac{b}{24}-\frac{1}{8}}\mathrm{hypergeom}\left(\left[\frac{b}{24}+\frac{1}{8}\,\frac{5}{8}+\frac{b}{24}\right]\,\left[1+\frac{b}{12}\right]\,-\frac{256x^3}{{\left(x-3\right)}^2\left(27x^2+14x+3\right)}\right)\right]$

(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).

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

For more information on Maple 2021 changes, see Updates in Maple 2021.