Thomae's Transformation

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Theorem

Let $a, b, c, e, f, s \in \C$.

Let $s = e + f - a - b - c$

Then:

$\ds \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {e, f} } \, \middle \vert \, 1} = \dfrac {\map \Gamma e \map \Gamma f \map \Gamma s } {\map \Gamma a \map \Gamma {s + b} \map \Gamma {s + c} } \map { {}_3 \operatorname F_2} { { {e - a, f - a, s} \atop {s + b, s + c} } \, \middle \vert \, 1} $


where:

$\ds \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {e, f} } \, \middle \vert \, 1}$ is the generalized hypergeometric function of $1$: $\ds \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} c^{\overline k} } { e^{\overline k} f^{\overline k} } \dfrac {1^k} {k!}$
$x^{\overline k}$ denotes the $k$th rising factorial power of $x$
$\map \Gamma {n + 1} = n!$ is the Gamma function.


Proof

First, we observe that:

\(\ds \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma e \map \Gamma f } \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {e, f} } \, \middle \vert \, 1}\) \(=\) \(\ds \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma e \map \Gamma f } \sum_{n \mathop = 0}^\infty \dfrac {\dfrac {\map \Gamma {a + n} } {\map \Gamma a} \dfrac {\map \Gamma {b + n} } {\map \Gamma b} \dfrac {\map \Gamma {c + n} } {\map \Gamma c} } {\dfrac {\map \Gamma {e + n} } {\map \Gamma e } \dfrac {\map \Gamma {f + n} } {\map \Gamma f } } \dfrac {1^n} {n!}\) Definition of Hypergeometric Function and Rising Factorial as Quotient of Factorials
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {e + n} \map \Gamma {f + n} } \times \dfrac {\map \Gamma {e + f - a + n} } {\map \Gamma {e + f - a + n} }\) multiplying by $1$ and $1^n = 1$
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {e + f - a + n} } \times \dfrac {\map \Gamma {e + f - a + n} \map \Gamma {a + n} } {\map \Gamma {e + n} \map \Gamma {f + n} }\) rearranging
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {e + f - a + n} } \times \dfrac {\map \Gamma {e + f - a + n} \map \Gamma {\paren {e + f - a + n} - \paren {e - a} - \paren {f - a} } } {\map \Gamma {\paren {e + f - a + n} - \paren {f - a} } \map \Gamma {\paren {e + f - a + n} - \paren {e - a} } }\) rearranging further
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {e + f - a + n} } \map F {e - a, f - a; e + f - a + n; 1}\) Gauss's Hypergeometric Theorem
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {e + f - a + n} } \sum_{m \mathop = 0}^\infty \dfrac {\dfrac {\map \Gamma {e - a + m} } {\map \Gamma {e - a } } \dfrac {\map \Gamma {f - a + m} } {\map \Gamma {f - a } } } {\dfrac {\map \Gamma {e + f - a + n + m} } {\map \Gamma {e + f - a + n } } } \dfrac {1^m} {m!}\) Definition of Hypergeometric Function and Rising Factorial as Quotient of Factorials
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \sum_{m \mathop = 0}^\infty \dfrac {\map \Gamma {b + n} \map \Gamma {c + n} \map \Gamma {e - a + m} \map \Gamma {f - a + m} } {m! n! \map \Gamma {e + f - a + n + m} \map \Gamma {e - a } \map \Gamma {f - a } }\) simplifying: $1^m = 1$ and $\map \Gamma {e + f - a + n}$ cancels
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \sum_{m \mathop = 0}^\infty \dfrac {\map \Gamma {b + n} \map \Gamma {c + n} \map \Gamma {e - a + m} \map \Gamma {f - a + m} } {m! n! \map \Gamma {e + f - a + n + m} \map \Gamma {e - a } \map \Gamma {f - a } } \times \dfrac {\map \Gamma b } {\map \Gamma b} \times \dfrac {\map \Gamma c } {\map \Gamma c} \times \dfrac {\map \Gamma {e + f - a + m} } {\map \Gamma {e + f - a + m} }\) multiplying by $1$
\(\ds \) \(=\) \(\ds \sum_{m \mathop = 0}^\infty \dfrac {\map \Gamma b \map \Gamma c \map \Gamma {e - a + m} \map \Gamma {f - a + m} } {m! \map \Gamma {e + f - a + m} \map \Gamma {e - a } \map \Gamma {f - a } } \sum_{n \mathop = 0}^\infty \dfrac {\dfrac {\map \Gamma {b + n} } {\map \Gamma b} \dfrac {\map \Gamma {c + n} } {\map \Gamma c} } {\dfrac {\map \Gamma {e + f - a + n + m} } {\map \Gamma {e + f - a + m} } } \dfrac {1^n} {n!}\) rearranging
\(\ds \) \(=\) \(\ds \sum_{m \mathop = 0}^\infty \dfrac {\map \Gamma b \map \Gamma c \map \Gamma {e - a + m} \map \Gamma {f - a + m} } {m! \map \Gamma {e + f - a + m} \map \Gamma {e - a } \map \Gamma {f - a } } \map F {b, c; e + f - a + m; 1}\) Definition of Hypergeometric Function and Rising Factorial as Quotient of Factorials
\(\ds \) \(=\) \(\ds \sum_{m \mathop = 0}^\infty \dfrac {\map \Gamma b \map \Gamma c \map \Gamma {e - a + m} \map \Gamma {f - a + m} } {m! \map \Gamma {e + f - a + m} \map \Gamma {e - a } \map \Gamma {f - a } } \dfrac {\map \Gamma {e + f - a + m} \map \Gamma {s + m} } {\map \Gamma {s + c + m} \map \Gamma {s + b + m} }\) Gauss's Hypergeometric Theorem and $s = e + f - a - b - c$
\(\ds \) \(=\) \(\ds \map \Gamma b \map \Gamma c \sum_{m \mathop = 0}^\infty \dfrac {\map \Gamma {e - a + m} \map \Gamma {f - a + m} } {m! \map \Gamma {e - a } \map \Gamma {f - a } } \dfrac {\map \Gamma {s + m} } {\map \Gamma {s + c + m} \map \Gamma {s + b + m} }\) rearranging and $\map \Gamma {e + f - a + m}$ cancels
\(\ds \) \(=\) \(\ds \map \Gamma b \map \Gamma c \sum_{m \mathop = 0}^\infty \dfrac {\map \Gamma {e - a + m} \map \Gamma {f - a + m} } {m! \map \Gamma {e - a } \map \Gamma {f - a } } \dfrac {\map \Gamma {s + m} } {\map \Gamma {s + c + m} \map \Gamma {s + b + m} } \times \dfrac {\map \Gamma s } {\map \Gamma s } \times \dfrac {\map \Gamma {s + b} } {\map \Gamma {s + b} } \times \dfrac {\map \Gamma {s + c} } {\map \Gamma {s + c} }\) multiplying by $1$
\(\ds \) \(=\) \(\ds \dfrac {\map \Gamma b \map \Gamma c \map \Gamma s } {\map \Gamma {s + b} \map \Gamma {s + c} } \sum_{m \mathop = 0}^\infty \dfrac {\dfrac {\map \Gamma {e - a + m} } {\map \Gamma {e - a } } \dfrac {\map \Gamma {f - a + m} } {\map \Gamma {f - a } } \dfrac {\map \Gamma {s + m} } {\map \Gamma s } } {\dfrac {\map \Gamma {s + b + m} } {\map \Gamma {s + b } } \dfrac {\map \Gamma {s + c + m} } {\map \Gamma {s + c } } } \dfrac {1^m} {m!}\) rearranging
\(\ds \) \(=\) \(\ds \dfrac {\map \Gamma b \map \Gamma c \map \Gamma s } {\map \Gamma {s + b} \map \Gamma {s + c} } \map { {}_3 \operatorname F_2} { { {e - a, f - a, s} \atop {s + b, s + c} } \, \middle \vert \, 1}\) Definition of Hypergeometric Function and Rising Factorial as Quotient of Factorials

We now have:

\(\ds \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma e \map \Gamma f } \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {e, f} } \, \middle \vert \, 1}\) \(=\) \(\ds \dfrac {\map \Gamma b \map \Gamma c \map \Gamma s } {\map \Gamma {s + b} \map \Gamma {s + c} } \map { {}_3 \operatorname F_2} { { {e - a, f - a, s} \atop {s + b, s + c} } \, \middle \vert \, 1}\)
\(\ds \leadsto \ \ \) \(\ds \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {e, f} } \, \middle \vert \, 1}\) \(=\) \(\ds \dfrac {\map \Gamma e \map \Gamma f \map \Gamma s } {\map \Gamma a \map \Gamma {s + b} \map \Gamma {s + c} } \map { {}_3 \operatorname F_2} { { {e - a, f - a, s} \atop {s + b, s + c} } \, \middle \vert \, 1}\)

$\blacksquare$


Also known as


Also see


Source of Name

This entry was named for Carl Johannes Thomae.


Sources

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