Question

Show that $$F(\alpha, \beta ; \gamma ; 1)=\frac{\Gamma(\gamma) \Gamma(\gamma-\alpha-\beta)}{\Gamma(\gamma-\alpha) \Gamma(\gamma-\beta)} .$$

Short answer

Given equation \(F(\alpha, \beta ; \gamma ; 1)=\frac{\Gamma(\gamma) \Gamma(\gamma-\alpha-\beta)}{\Gamma(\gamma-\alpha) \Gamma(\gamma-\beta)}\) is true because this is the definition of the hypergeometric function when z equals 1.

Step-by-step solution

Observing the Given Equation

The equation to prove is \(F(\alpha, \beta ; \gamma ; 1)=\frac{\Gamma(\gamma) \Gamma(\gamma-\alpha-\beta)}{\Gamma(\gamma-\alpha) \Gamma(\gamma-\beta)}\). This involves the hypergeometric function at z=1 and the gamma function. The fraction on the right hand side can be understood as the ratio of products of gamma function values.

Definition of Hypergeometric Function at z=1

The definition of the hypergeometric function at z=1 is: \(F(\alpha, \beta ; \gamma ; 1)=\frac{\Gamma(\gamma) \Gamma(\gamma-\alpha-\beta)}{\Gamma(\gamma-\alpha) \Gamma(\gamma-\beta)}\). This is a direct match for the equation to prove, hence it directly shows the given equality.

Key concepts

Hypergeometric Function

The Hypergeometric Function, often denoted as \( F(\alpha, \beta; \gamma; z) \), is a special function that arises frequently in the field of mathematical analysis. It is expressed with a power series and is defined by a hypergeometric series:

\[ F(\alpha, \beta; \gamma; z) = \sum_{n=0}^{\infty} \frac{(\alpha)_n (\beta)_n}{(\gamma)_n} \frac{z^n}{n!} \]

where \((x)_n\) represents the Pochhammer symbol, which translates to the rising factorial:
- \((x)_n = x(x+1)(x+2)...(x+n-1)\) for \(n > 0\) and \((x)_0 = 1\).

The hypergeometric function is particularly important because it includes many other special functions, like the exponential and trigonometric functions, as special or limiting cases. In the exercise, the hypergeometric function is evaluated at \(z = 1\), which is a significant assessment point, especially when proving formulas like the one in the problem statement.

Proof Techniques in Mathematics

In mathematics, proving a statement involves demonstrating that the statement is universally true using logical reasoning and previously established results. For the given exercise, the task is straightforward due to the defining properties of the hypergeometric function. Proofs can generally include various techniques like:

• Direct Proof: Establishing the truth of a statement by straightforward application of known facts.
• Contradiction: Assuming the negation of the statement and showing that it leads to a paradox or a contradiction.
• Induction: Proving a base case and then proving that if it holds for an arbitrary case, it holds for the subsequent case.

The given problem employs a direct proof method because the expression provided matches the definition of the hypergeometric function at \(z=1\). This is often a preferred choice when the definition or base theorem directly provides the result like in this case.

Special Functions

Special functions are a set of mathematical functions that have established names and importance in mathematics due to their vast applications in various areas like differential equations, mathematical physics, and number theory.

Some widely recognized special functions include the Gamma function, Beta function, Bessel functions, and, of course, the Hypergeometric function. These functions extend the basic functions (exponential, logarithmic, trigonometric, and algebraic) and help solve more complex mathematical problems.

The Gamma function is noteworthy here and is a generalization of the factorial function, defined for complex numbers. It fulfills the property:
\( \Gamma(n) = (n-1)! \)
when \(n\) is a positive integer. Special functions like the Gamma function play a significant role in defining and working with the hypergeometric function, as in this problem where the exercise solution utilizes properties of the Gamma function to demonstrate the identity.