Gauss's Hypergeometric Theorem

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Theorem

Let $a, b, c \in \C$.

Let $c \notin \Z_{\le 0}$.

Let $\map \Re {c - a - b} > 0$.


Then:

$\map F {a, b; c; 1} = \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }$

where:

$\map F {a, b; c; 1}$ is the Gaussian hypergeometric function of $1$: $\ds \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} } { c^{\overline k} } \dfrac {1^k} {k!}$
$x^{\overline k}$ denotes the $k$th rising factorial power of $x$
$\map \Gamma {n + 1} = n!$ is the Gamma function.


Corollary 1

Let $\map \Re {1 - a} > 0$.

Let $c \notin \Z_{\le 0}$ and $c \ne 1$.

Then:

$\ds \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {\paren {c - 1 + k} k!} = \dfrac {\map \Gamma {c - 1} \map \Gamma {1 - a} } {\map \Gamma {c - a} }$


Corollary 2

Let $\map \Re {a - 1} < 0$.

Then:

$\ds \dfrac 1 a + \dfrac {a } {\paren {a + 1} 1!} + \dfrac {a \paren {a + 1} } {\paren {a + 2} 2!} + \dfrac {a \paren {a + 1} \paren {a + 2} } {\paren {a + 3} 3!} + \cdots = \dfrac {\pi} {\map \sin {\pi a } } $


Proof 1

Let $x, y, n \in \C$ be complex numbers such that $\map \Re {x + y + n + 1} > 0$.

Let $u \in \C$ be a complex number such that $\cmod u < 1$.

Expanding the product of $\paren {1 + u}^{y + n}$ and $\paren {\dfrac {1 + u} u}^x$:

\(\ds \paren {1 + u}^{y + n}\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \binom {y + n} k u^k\) Binomial Theorem - Complex Numbers
\(\ds \paren {1 + \dfrac 1 u}^x\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \binom x k u^{-k}\) Binomial Theorem - Complex Numbers, Power of Product
\(\ds \leadsto \ \ \) \(\ds \paren {1 + u}^{y + n} \paren {\dfrac {1 + u} u}^x\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \binom {y + n} k u^k \sum_{k \mathop = 0}^\infty \binom x k u^{-k}\) multiplying

The coefficient $a_n$ of $u^n$ of the product $\dfrac {\paren {1 + u}^{x + y + n} } {u^x}$ above can be determined by setting $k = k + n$ in the series with $\dbinom {y + n} k$:

\(\ds a_n\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \binom {y + n} {k + n} \binom x k\) substituting $k = k + n$ in first series: $u^n = u^{k+n} u^{-k}$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {y + n}!} {\paren {k + n}! \paren {y - k}!} \dfrac {x!} {k! \paren {x - k}!}\) Definition of Binomial Coefficient
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {y + n}!} {\paren {k + n}! \paren {y - k}!} \dfrac {x!} {k! \paren {x - k}!} \paren {\dfrac {n! y!} {n! y!} }\) multiplying by $1$
\(\ds \) \(=\) \(\ds \dfrac {\paren {y + n}! } {n! y!} \sum_{k \mathop = 0}^\infty \dfrac {n! } {k! \paren {k + n}! } \dfrac {y!} {\paren {y - k}!} \dfrac {x!} {\paren {x - k}!}\) moving $\dfrac {\paren {y + n}! } {n! y!}$ outside the sum and rearranging
\(\ds \) \(=\) \(\ds \dfrac {\map \Gamma {y + n + 1} } {\map \Gamma {n + 1} \map \Gamma {y + 1} } \sum_{k \mathop = 0}^\infty \dfrac {n! } {k! \paren {k + n}! } \dfrac {y!} {\paren {y - k}!} \dfrac {k!} {k!} \dfrac {x!} {\paren {x - k}!} \dfrac {k!} {k!}\) multiplying by $1$
\(\ds \) \(=\) \(\ds \dfrac {\map \Gamma {y + n + 1} } {\map \Gamma {n + 1} \map \Gamma {y + 1} } \sum_{k \mathop = 0}^\infty \dfrac {n! } {k! \paren {k + n}! } \dbinom y k \dbinom x k \paren {k!}^2\) Definition of Binomial Coefficient
\(\ds \) \(=\) \(\ds \dfrac {\map \Gamma {y + n + 1} } {\map \Gamma {n + 1} \map \Gamma {y + 1} } \sum_{k \mathop = 0}^\infty \dfrac {n! } {k! \paren {k + n}! } \paren {-1}^k \dbinom {-y + k - 1} k \paren {-1}^k \dbinom {-x + k - 1} k \paren {k!}^2\) Negated Upper Index of Binomial Coefficient
\(\ds \) \(=\) \(\ds \dfrac {\map \Gamma {y + n + 1} } {\map \Gamma {n + 1} \map \Gamma {y + 1} } \sum_{k \mathop = 0}^\infty \dfrac {n! } {k! \paren {k + n}! } \dfrac {\paren {-y + k - 1}!} {k! \paren {-y - 1}!} \dfrac {\paren {-x + k - 1}!} {k! \paren {-x - 1}!} \paren {k!}^2\) Definition of Binomial Coefficient, $\paren {-1}^{2 k} = 1$
\(\ds \) \(=\) \(\ds \dfrac {\map \Gamma {y + n + 1} } {\map \Gamma {n + 1} \map \Gamma {y + 1} } \sum_{k \mathop = 0}^\infty \dfrac { \paren {-x}^{\overline k} \paren {-y}^{\overline k} } { k! \paren {n + 1}^{\overline k} }\) Rising Factorial as Quotient of Factorials, and $k!$ cancels


Now expand $\paren {1 + u}^{x + y + n}$ and divide by $u^x$:

\(\ds \dfrac {\paren {1 + u}^{x + y + n} } {u^x}\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \binom {x + y + n} k u^{k - x}\) Binomial Theorem - Complex Numbers


The coefficient $a_n$ of $u^n$ of the product $\dfrac {\paren {1 + u}^{x + y + n} } {u^x}$ above is:

\(\ds a_n\) \(=\) \(\ds \binom {x + y + n} {x + n}\) $n = k - x$, so $k = x + n$
\(\ds \) \(=\) \(\ds \dfrac {\map \Gamma {x + y + n + 1} } {\map \Gamma {x + n + 1} \map \Gamma {y + 1} }\) Definition of Binomial Coefficient


Equating coefficients gives us:

\(\ds \dfrac {\map \Gamma {y + n + 1} } {\map \Gamma {n + 1} \map \Gamma {y + 1} } \sum_{k \mathop = 0}^\infty \dfrac { \paren {-x}^{\overline k} \paren {-y}^{\overline k} } {\paren {n + 1}^{\overline k} } \dfrac 1 {k!}\) \(=\) \(\ds \dfrac {\map \Gamma {x + y + n + 1} } {\map \Gamma {x + n + 1} \map \Gamma {y + 1} }\)
\(\ds \leadsto \ \ \) \(\ds \sum_{k \mathop = 0}^\infty \dfrac { \paren {-x}^{\overline k} \paren {-y}^{\overline k} } { \paren {n + 1}^{\overline k} } \dfrac 1 {k!}\) \(=\) \(\ds \dfrac {\map \Gamma {x + y + n + 1} \map \Gamma {n + 1} } {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} }\) dividing both sides by $\dfrac {\map \Gamma {y + n + 1} } {\map \Gamma {n + 1} \map \Gamma {y + 1} }$
\(\ds \leadsto \ \ \) \(\ds \map F {-x, -y; n + 1; 1}\) \(=\) \(\ds \dfrac {\map \Gamma {x + y + n + 1} \map \Gamma {n + 1} } {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} }\) Definition of Hypergeometric Function

Letting $a = -x$, $b = -y$ and $c = n + 1$, we obtain:

\(\ds \map F {a, b; c; 1}\) \(=\) \(\ds \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }\)

$\blacksquare$


Proof 2

From Euler's Integral Representation of Hypergeometric Function, we have:

$\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - x t}^{- a} \rd t$

Where $a, b, c \in \C$.

and $\size x < 1$

and $\map \Re c > \map \Re b > 0$.

Since Euler's Integral Representation only applies where $\size x < 1$, we will determine the limit of the integral as $x \to 1$.


Therefore:

\(\ds \map F {a, b; c; x}\) \(=\) \(\ds \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - \paren {1} t}^{- a} \rd t\)
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - a - 1} \rd t\) simplifying and Product of Powers
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \dfrac {\map \Gamma b \map \Gamma {c - a - b } } {\map \Gamma {c - a } }\) Definition of Beta Function
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }\) simplifying and canceling $\map \Gamma b$


$\blacksquare$


Examples

Example: $\map F {1, 2; 4; 1}$

$1 + \dfrac 2 4 + \paren {\dfrac {2 \times 3} {4 \times 5} } + \paren {\dfrac {2 \times 3 \times 4} {4 \times 5 \times 6} } + \cdots = 3$


Example: $\map F {1, 1; \dfrac 5 2; 1}$

$1 + \dfrac 2 5 + \paren {\dfrac {2 \times 4} {5 \times 7} } + \paren {\dfrac {2 \times 4 \times 6} {5 \times 7 \times 9} } + \cdots = 3$


Example: $\map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; 1}$

$1 + \paren {\dfrac 1 {2^1 \times 3 \times 1!} } + \paren {\dfrac {1 \times 3} {2^2 \times 5 \times 2!} } + \paren {\dfrac {1 \times 3 \times 5} {2^3 \times 7 \times 3!} } + \cdots = \dfrac \pi 2$


Example: $\dfrac {10} 3 \map F {\dfrac 3 {10}, \dfrac 3 {10}; \dfrac {13} {10}; 1}$

$\paren {\dfrac 1 {10^{-1} \times 3 \times 0!} } + \paren {\dfrac 3 {10^0 \times 13 \times 1!} } + \paren {\dfrac {3 \times 13} {10^1 \times 23 \times 2!} } + \paren {\dfrac {3 \times 13 \times 23} {10^2 \times 33 \times 3!} } + \cdots = \dfrac {2 \pi} \phi$


Also see


Source of Name

This entry was named for Carl Friedrich Gauss.


Sources

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