Hypergeometric DE's & the Riemann P-function

In summary, the conversation discusses the process of deriving a second linearly independent solution to a hypergeometric differential equation. The solution is found using manipulations of the Riemann P-function, which lists the exponents of each solution at the singular points 0, 1, and infinity. The derivation involves taking out a factor of z^{1-\gamma} and adjusting the parameters to match the exponents at each singular point. While there are other ways to find the second solution, this method is considered clever and important to understand.
  • #1
Kirjava
27
1
Apologies in advance for the TeX in this post, I'm new and having difficulty with the formatting.

Homework Statement


I'm trying to understand the logic my professor uses to derive a second linearly independent solution to the hypergeometric DE:
[tex]z(1-z)\frac{d^2 w}{dz^2} + (\gamma - z(1+\alpha + \beta)\frac{dw}{dz} -\alpha\beta w = 0 [/tex].
It is given that the solutions have exponents at each of the singularities (0,1,[tex]\infty[/tex]) which are listed in the Riemann P-function:
[tex]
P\left(
\begin{array}{ccc}
0 & 1 & \infty\\
0 & 0 & \alpha\\
1-\gamma & \gamma -\alpha -\beta & \beta
\end{array} \right) [/tex].
The hypergeometric series [tex]$ _2F_1 \left( \alpha , \beta ; \gamma ; z\right)$[/tex] is the first solution with exponents given in the second row. To "derive" an expression for the second solution whose exponents appear in the last row, the following manipulations are made:

[tex]
$ y = P\left(
\begin{array}{ccc}
0 & 1 & \infty\\
0 & 0 & \alpha\\
1-\gamma & \gamma -\alpha -\beta & \beta
\end{array} \right) = z^{1-\gamma} P\left(
\begin{array}{ccc}
0 & 1 & \infty\\
\gamma -1 & 0 & \alpha+1-\gamma\\
0 & \gamma -\alpha -\beta & \beta+1-\gamma
\end{array} \right)$ [/tex]
and therefore (apparently) [tex] z^{1-\gamma} \, _2F_1 \left(1+ \alpha-\gamma , 1+\beta-\gamma ; 2-\gamma ; z\right)[/tex] is the second linearly independent solution.

Homework Equations



All above

The Attempt at a Solution



These manipulations seem dishonest to me (i.e. I don't understand them). If the Riemann P-function is just a table for summarizing each solution's exponents at the singular points (0,1,[tex]\infty[/tex]), then I'm not even sure what the statement y = P(...) could mean. I can certainly see that if we have a solution w with exponents given in a particular row, then taking out a factor [tex]z^{1-\gamma}[/tex] leaves us with [tex]wz^{\gamma-1}[/tex], which will have exponents given in the respective row of the second P function above. Now the "derivation" seems to assume that [tex]wz^{\gamma-1}[/tex] is the solution to a hypergeometric equation with parameters adjusted to match its exponents at each of the singular points. I don't see how this is warranted.

I'm aware that there are other more direct ways of deriving the second solution (if the DE is tackled directly using a Frobenius series the two solutions pop right out). However I'd rather like to understand these manipulations since they seem rather clever, and we're expected to understand and use many others of a similar kind.
 
Last edited:
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  • #2
A:It's a basic fact that the exponents of a solution at the points $0, 1, \infty$ are completely determined by its hypergeometric coefficients. In other words, any two solutions to a hypergeometric equation whose exponents are given by two different rows of the P-symbol must have different hypergeometric coefficients. That's why the professor is manipulating the coefficients of the hypergeometric function to find the second solution. He doesn't have to solve the equation again to find it; he just uses the fact that these coefficients determine the exponents.
 

Related to Hypergeometric DE's & the Riemann P-function

1. What is the Riemann P-function?

The Riemann P-function is a mathematical function that is closely related to the hypergeometric differential equation. It is defined as the derivative of the hypergeometric function with respect to its variable.

2. How are hypergeometric differential equations related to the Riemann P-function?

Hypergeometric differential equations are closely related to the Riemann P-function because they can be solved using the Riemann P-function. In fact, the Riemann P-function is a fundamental solution to the hypergeometric differential equation.

3. What is the significance of the Riemann P-function in mathematics?

The Riemann P-function is significant in mathematics because it has many important applications in various fields such as number theory, algebraic geometry, and mathematical physics. It is also closely related to other important mathematical functions such as the gamma function and the elliptic functions.

4. How is the hypergeometric differential equation used in real-world problems?

The hypergeometric differential equation is used in various real-world problems, particularly in physics and engineering. It is used to model a wide range of physical phenomena, including heat transfer, fluid dynamics, and quantum mechanics.

5. Are there any practical applications of the Riemann P-function?

Yes, there are many practical applications of the Riemann P-function. Some examples include its use in cryptography for secure communication, its role in number theory for solving certain types of equations, and its application in engineering for solving differential equations that arise in various fields.

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