# 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

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

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

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

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 ${\ displaystyle C_ {1}, C_ {2} \ in \ mathbb {C}}$

## 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.