Pfaff's Transformation

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Theorem

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

Let $\size x < \dfrac 1 2$

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

Then:

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


where:

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


Proof

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

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

Letting $s = \paren {1 - t}$, we now have:

\(\ds \map F {a, b; c; x}\) \(=\) \(\ds \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_1^0 \paren {1 - s}^{b - 1} s^{c - b - 1} \paren {1 - x\paren {1 - s} }^{- a} \paren {-\rd s}\) substituting $s = \paren {1 - t}$
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 \paren {1 - s}^{b - 1} s^{c - b - 1} \paren {1 - x\paren {1 - s} }^{- a} \rd s\) reversing limits of integration
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 \paren {1 - s}^{b - 1} s^{c - b - 1} \paren {1 - x + xs }^{- a} \rd s\) simplifying
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 \paren {1 - s}^{b - 1} s^{c - b - 1} \paren {1 - x + xs }^{- a} \rd s \times \dfrac {\paren {1 - x}^{-a} } {\paren {1 - x}^{-a} }\) multiplying by $1$
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \dfrac {\map \Gamma c \paren {1 - x}^{-a} } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 \paren {1 - s}^{b - 1} s^{c - b - 1} \paren {1 + \dfrac {xs} {1 - x} }^{- a} \rd s\) rearranging
\(\text {(1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \dfrac {\map \Gamma c \paren {1 - x}^{-a} } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 \paren {1 - s}^{b - 1} s^{c - b - 1} \paren {1 - \dfrac {xs} {x - 1} }^{- a} \rd s\) switching the sign in the last term


Letting $\size x < \dfrac 1 2$ and expanding the product of $\paren {1 - \dfrac {xs} {x - 1} }^{-a}$:

\(\ds \paren {1 - \dfrac {xs} {x - 1} }^{-a}\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \binom {-a} k \paren {-1}^k \paren {\dfrac {xs} {x - 1} }^k\) Binomial Theorem - Complex Numbers
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \paren {\binom {a + k - 1} k \paren {-1}^k} \paren {-1}^k \paren {\dfrac {xs} {x - 1} }^k\) Negated Upper Index of Binomial Coefficient
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \binom {a + k - 1} k \paren {\dfrac {xs} {x - 1} }^k\) $\paren {-1}^{2k} = 1$
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {a + k - 1}! } {k! \paren {a - 1}! } \paren {\dfrac x {\paren {x - 1} } }^k s^k\) Definition of Binomial Coefficient
\(\text {(2)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {k!} \paren {\dfrac x {\paren {x - 1} } }^k s^k\) Rising Factorial as Quotient of Factorials


Therefore:

\(\ds \dfrac {\map \Gamma c \paren {1 - x}^{-a} } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 \paren {1 - s}^{b - 1} s^{c - b - 1} \paren {1 - \dfrac {xs} {x - 1} }^{- a} \rd s\) \(=\) \(\ds \dfrac {\map \Gamma c \paren {1 - x}^{-a} } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 \paren {1 - s}^{b - 1} s^{c - b - 1} \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {k!} \paren {\dfrac x {\paren {x - 1} } }^k s^k \rd s\) substituting $\paren {2}$ into $\paren {1}$
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \dfrac {\map \Gamma c \paren {1 - x}^{-a} } {\map \Gamma b \map \Gamma {c - b} } \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {k!} \paren {\dfrac x {\paren {x - 1} } }^k \int_0^1 \paren {1 - s}^{b - 1} s^{c - b - 1} s^k \rd s\)
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \dfrac {\map \Gamma c \paren {1 - x}^{-a} } {\map \Gamma b \map \Gamma {c - b} } \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {k!} \paren {\dfrac x {\paren {x - 1} } }^k \int_0^1 \paren {1 - s}^{b - 1} s^{k + c - b - 1} \rd s\) Product of Powers
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \dfrac {\map \Gamma c \paren {1 - x}^{-a} } {\map \Gamma b \map \Gamma {c - b} } \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } {k!} \paren {\dfrac x {\paren {x - 1} } }^k \dfrac {\map \Gamma {k + c - b} \map \Gamma b } {\map \Gamma {k + c} }\) Definition of Beta Function
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \paren {1 - x}^{-a} \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} \paren {c - b}^{\overline k} } { k! c^{\overline k} } \paren {\dfrac x {\paren {x - 1} } }^k\) Rising Factorial as Quotient of Factorials and $\map \Gamma b$ cancels
\(\ds \leadsto \ \ \) \(\ds \) \(=\) \(\ds \paren {1 - x}^{-a} \map F {a, c - b; c; \dfrac x {x - 1} }\) Definition of Gaussian Hypergeometric Function


Therefore:

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

$\blacksquare$


Also see


Source of Name

This entry was named for Johann Friedrich Pfaff.


Sources

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