User:EGM6321.f12.team4.Rui.C/Homework5 R5.12

Statement
Given a hypergeometric differential equation:

(1) Verify whether it is exact;

(2) Verify whether it is in the power form:


 * where

(3)Verify that $$F(a,b;c;x)$$ in (1) p.64-4 as shown below is indeed a solution of (5.12.0.1):


 * where Pochhammer's symbol is denoted as:

Solution (1)
First exactness condition:

Equation (5.12.0.1) can be written in the form:

where

Then it is obvious that the first exactness condition holds.

For the second exactness condition:

The partial derivatives in these two equations can be derived as:

Since $$f$$ is only a function of x,

On the other hand,

Thus,

Substituting (5.12.1.6)~(5.12.1.12) into (5.12.1.3) and (5.12.1.4), these two relations of the second exactness condition can be written as:

Then the first relation does not hold. So the second exactness condition does not hold and equation (5.12.0.1) is not exact.

Solution (2)
By comparing (5.12.0.1) and (5.12.0.2), it is clear that the term $$f(x)=x(1-x)$$ can not be written in the form of $$\alpha x^r$$, because there are two subterms in the former expression with different exponents while there is only one exponent in the latter term.

Similarly, $$[c-(a+b+1)x]y'$$ cannot also be written in the form of $$\beta x^sy'$$.

Although the term $$-aby$$ can be written as $$\gamma x^ty$$ if we let $$\gamma=-ab$$ and $$t=0$$, the overall equation is not in power form.

Solution (3)
Assume

as is shown in (5.12.0.3).

Then we can substitute it into the left part of equation (5.12.0.1):

Here the derivative for k=0,1 in the first term equal to 0, since they are the derivative of a constant:

For k=0:

For k=1:

Similarly the second term in (5.12.3.2) for k=0 is also equal to 0.

Then we can further reorganize (5.12.3.2) as:

To verify that this expression equals to zero, each coefficient of the linearly independent polynomial terms($$1,x,x^2,x^3,\cdots$$) should equal to zero. In order to evaluate these coefficients, we shall first make a index transformation for the first and the third terms of (5.12.3.5):

Then (5.12.3.5) becomes:

Then it can be further written as:

Because of the derivative process, the first and the fourth terms exists for only k>0 (same for k'), while second term exists for only k>1. Thus we shall examine k=0, k=1 and k>1 separately.

For k=0, namely for the coefficient of polynomial term '1':

For k=1, namely for the coefficient of polynomial term 'x':

For any term k>1, a general coefficient form for each of the $$x^k$$ polynomial from (5.12.3.8):

It can be further reorganized as:

Thus, any coefficients for the polynomial terms are equal to 0. Then the overall expression for the left part of (5.12.0.1), namely the expression (5.12.3.2) equals to 0.

Then we can conclude that the expression (5.12.0.3) is a solution of the equation (5.12.0.1).

Author and References

 * Solved and Typed by -- Rui Che