Hypergeometric differential equation

In 1801 Leonhard Euler gave the solution to the hypergeometric differential equation. It is closely related to the Gaussian hypergeometric function , which was first systematically investigated by Carl Friedrich Gauß .

Hypergeometric differential equation

The hypergeometric function , where the gamma function denotes, satisfies the linear differential equation of the 2nd order : ${\displaystyle \textstyle {}_{2}F_{1}(a,b;c;z)=\sum _{k=0}^{\infty }{\frac {\Gamma (a+k)\,\Gamma (b+k)\,\Gamma (c)}{\Gamma (a)\,\Gamma (b)\,\Gamma (c+k)}}{\frac {z^{k}}{k!}}}$ ${\displaystyle \Gamma (\cdot )}$

${\displaystyle z(1-z){\frac {\rm {d^{2}}}{{\rm {d}}z^{2}}}\;{}_{2}F_{1}(a,b;c;z)+\left[c-(a+b+1)z\right]{\frac {\rm {d}}{{\rm {d}}z}}\;{}_{2}F_{1}(a,b;c;z)-ab\;{}_{2}F_{1}(a,b;c;z)=0}$.

Singularities

The differential equation of the 2nd order has three liftable singularities , the values ​​of which are determined in the following.

Based on the hypergeometric differential equation in the illustration

${\displaystyle {\frac {\rm {d^{2}}}{{\rm {d}}z^{2}}}\;{}_{2}F_{1}(a,b;c;z)+p(z){\frac {\rm {d}}{{\rm {d}}z}}\;{}_{2}F_{1}(a,b;c;z)-q(z)\;{}_{2}F_{1}(a,b;c;z)=0}$

With

${\displaystyle p(z)={\frac {c-(a+b+1)z}{z(1-z)}}={\frac {c-cz+(c-a-b-1)z}{z(1-z)}}={\frac {c}{z}}+{\frac {c-a-b-1}{1-z}}}$

and

${\displaystyle q(z)={\frac {ab}{z(1-z)}}}$

we get the two liftable singularities at and . ${\displaystyle z=0}$ ${\displaystyle z=1}$

The third liftable singularity is obtained through the substitution . First, the derivatives of the hypergeometric function are substituted as follows: ${\displaystyle \textstyle t={\frac {1}{z}},{\frac {{\rm {d}}t}{{\rm {d}}z}}={\frac {-1}{z^{2}}}=-t^{2}}$

${\displaystyle {\frac {\rm {d}}{{\rm {d}}z}}\;{}_{2}F_{1}(a,b;c;z)={\frac {\rm {d}}{{\rm {d}}t}}\;{}_{2}F_{1}(a,b;c;t)\cdot {\frac {{\rm {d}}t}{{\rm {d}}z}}=-t^{2}\cdot {\frac {\rm {d}}{{\rm {d}}t}}\;{}_{2}F_{1}(a,b;c;t)}$

and

${\displaystyle {\begin{aligned}{\frac {\rm {d^{2}}}{{\rm {d}}z^{2}}}\;{}_{2}F_{1}(a,b;c;z)&={\frac {\rm {d}}{{\rm {d}}t}}{\Big (}-t^{2}\cdot {\frac {\rm {d}}{{\rm {d}}t}}\;{}_{2}F_{1}(a,b;c;t){\Big )}\cdot {\frac {{\rm {d}}t}{{\rm {d}}z}}\\&=-t^{2}{\Big (}-2t\cdot {\frac {\rm {d}}{{\rm {d}}t}}\;{}_{2}F_{1}(a,b;c;t)-t^{2}\cdot {\frac {\rm {d^{2}}}{{\rm {d}}t^{2}}}\;{}_{2}F_{1}(a,b;c;t){\Big )}\\&=t^{4}\cdot {\frac {\rm {d^{2}}}{{\rm {d}}t^{2}}}\;{}_{2}F_{1}(a,b;c;t)+2t^{3}\cdot {\frac {\rm {d}}{{\rm {d}}t}}\;{}_{2}F_{1}(a,b;c;t)\end{aligned}}}$

Thus the hypergeometric differential equation, after division by , takes on the following form: ${\displaystyle t^{4}}$

${\displaystyle {\frac {\rm {d^{2}}}{{\rm {d}}t^{2}}}\;{}_{2}F_{1}(a,b;c;t)+{\tilde {p}}(t)\cdot {\frac {\rm {d}}{{\rm {d}}t}}\;{}_{2}F_{1}(a,b;c;t)-{\tilde {q}}(t)\;{}_{2}F_{1}(a,b;c;t)=0}$

With

${\displaystyle {\tilde {p}}(t)={\frac {2}{t}}+{\frac {1}{t^{2}}}p(z={\tfrac {1}{t}})={\frac {2}{t}}+{\frac {1}{t^{2}}}{\Big (}ct+{\frac {c-a-b-1}{1-{\frac {1}{t}}}}{\Big )}={\frac {c+2}{t}}+{\frac {c-a-b-1}{t(t-1)}}}$

and

${\displaystyle {\tilde {q}}(t)={\frac {1}{t^{4}}}q(z={\tfrac {1}{t}})={\frac {1}{t^{4}}}{\frac {ab}{{\frac {1}{t}}(1-{\frac {1}{t}})}}={\frac {ab}{t^{2}(t-1)}}}$

Accordingly, the hypergeometric differential equation also has a lifting singularity at. ${\displaystyle z={\tfrac {1}{t}}=\infty }$

Solution of the hypergeometric differential equation

With the power series approach with complex coefficients , the hypergeometric differential equation reads: ${\displaystyle \textstyle u(z)=\sum _{k=0}^{\infty }u_{k}z^{k}}$ ${\displaystyle u_{k}}$

${\displaystyle z(1-z){\frac {\rm {d^{2}}}{{\rm {d}}z^{2}}}\sum _{k=0}^{\infty }u_{k}z^{k}+\left[c-(a+b+1)z\right]{\frac {\rm {d}}{{\rm {d}}z}}\sum _{k=0}^{\infty }u_{k}z^{k}-ab\sum _{k=0}^{\infty }u_{k}z^{k}=0}$.

After executing the derivations, the representation results

${\displaystyle z(1-z)\sum _{k=2}^{\infty }k(k-1)u_{k}z^{k-2}+\left[c-(a+b+1)z\right]\sum _{k=1}^{\infty }ku_{k}z^{k-1}-ab\sum _{k=0}^{\infty }u_{k}z^{k}=0}$.

Summarizing the powers of leads to ${\displaystyle z}$

${\displaystyle \sum _{k=2}^{\infty }k(k-1)u_{k}z^{k-1}-\sum _{k=2}^{\infty }k(k-1)u_{k}z^{k}+c\sum _{k=1}^{\infty }ku_{k}z^{k-1}-(a+b+1)\sum _{k=1}^{\infty }ku_{k}z^{k}-ab\sum _{k=0}^{\infty }u_{k}z^{k}=0}$.

The index shift results in

${\displaystyle \sum _{k=0}^{\infty }(k+1)ku_{k+1}z^{k}-\sum _{k=0}^{\infty }k(k-1)u_{k}z^{k}+c\sum _{k=0}^{\infty }(k+1)u_{k+1}z^{k}-(a+b+1)\sum _{k=0}^{\infty }ku_{k}z^{k}-ab\sum _{k=0}^{\infty }u_{k}z^{k}=0}$.

This equation is obviously true if:

${\displaystyle (k+1)ku_{k+1}-k(k-1)u_{k}+c(k+1)u_{k+1}-(a+b+1)ku_{k}-abu_{k}=0}$.

The following recursion has thus been found for the coefficient : ${\displaystyle u_{k}}$

${\displaystyle {\begin{aligned}u_{k+1}&={\frac {k(k-1)+(a+b+1)k+ab}{(k+1)k+c(k+1)}}u_{k}\qquad ={\frac {k^{2}-k+ka+kb+k+ab}{(c+k)(1+k)}}u_{k}\\&={\frac {k^{2}+ka+kb+ab}{(c+k)(1+k)}}u_{k}\qquad \qquad \qquad \;\;={\frac {(a+k)(b+k)}{(c+k)(1+k)}}u_{k}&={\frac {(a,k)(b,k)}{(c,k)(1,k)}}u_{k}\end{aligned}}}$

Here the Pochhammer symbol denotes . ${\displaystyle (x,n)\equiv {\tfrac {\Gamma (x+n)}{\Gamma (x)}}}$

If the initial value is set, the first basic solution of the hypergeometric differential equation is: ${\displaystyle u_{0}=1}$

${\displaystyle u(z)={}_{2}F_{1}(a,b;c;z)=\sum _{k=0}^{\infty }{\frac {(a,k)(b,k)}{(c,k)(1,k)}}z^{k}=\sum _{k=0}^{\infty }{\frac {\Gamma (a+k)\,\Gamma (b+k)\,\Gamma (c)}{\Gamma (a)\,\Gamma (b)\,\Gamma (c+k)}}{\frac {z^{k}}{k!}}}$.

For is obtained as a second linearly independent basic solution ${\displaystyle c\notin \mathbb {Z} }$

${\displaystyle v(z)=z^{1-c}{}_{2}F_{1}(a-c+1,b-c+1;2-c;z)}$

Both together span the entire solution space of the hypergeometric differential equation:

${\displaystyle y(z)=C_{1}u(z)+C_{2}v(z)}$ With ${\displaystyle C_{1},C_{2}\in \mathbb {C} }$

See also

• Gaussian hypergeometric function
• Gamma function

literature

• Leonhard Euler : Specimen transformationis singularis serierum . In: Nova Acta Academiae Scientarum Imperialis Petropolitinae . 12, 1801, pp. 58-70.

• Ludwig Bieberbach : Theory of Differential Equations Springer, Berlin 1930, Second Section, Chapter IV, § 7, uni-goettingen.de

Individual evidence

1. ^ Leonhard Euler: Transformationis Singularis, Nova Acta Academiae Scientiarum Imperialis Petropolitanae, Volume 12, 1801, Pages 58-70, online at books.google.de
2. ↑ Erwin Kreyszig: Advanced Engineering Mathematics, John Wiley & Sons 1988, page 204f.