Euler's Transformation
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Theorem
Let $a, b, c \in \C$.
Let $\size x < 1$
Let $\map \Re c > \map \Re b > 0$.
Then:
- $\ds \map F {a, b; c; x} = \paren {1 - x}^{c - a - b} \map F {c - a, c - b; c; x}$
where $\map F {a, b; c; x}$ is the Gaussian hypergeometric function of $x$.
Proof
First, we observe:
\(\ds \dfrac {\dfrac x {x - 1} } {\dfrac x {x - 1} - 1}\) | \(=\) | \(\ds \dfrac {\dfrac x {x - 1} } {\dfrac x {x - 1} -1} \times \dfrac {\paren {x - 1} } {\paren {x - 1} }\) | multiplying by $1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac x {x - \paren {x - 1} }\) | ||||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds x\) |
Applying Pfaff's Transformation twice, we obtain:
\(\ds \map F {a, b; c; x}\) | \(=\) | \(\ds \paren {1 - x}^{-a} \map F {a, c - b; c; \dfrac x {x - 1} }\) | First application of Pfaff's Transformation | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {1 - x}^{-a} \map F {c - b, a; c; \dfrac x {x - 1} }\) | symmetry in first two terms | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {1 - x}^{-a} \paren {1 - x}^{c - b} \map F {c - b, c - a; c; \dfrac {\dfrac x {x - 1} } {\dfrac x {x - 1} - 1} }\) | Second application of Pfaff's Transformation | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {1 - x}^{-a} \paren {1 - x}^{c - b} \map F {c - a, c - b; c; x}\) | symmetry in first two terms and $(1)$ above | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {1 - x}^{c - a - b} \map F {c - a, c - b; c; x}\) | Product of Powers |
Therefore, after two applications of Pfaff's Transformation, we have:
- $\map F {a, b; c; x} = \paren {1 - x}^{c - a - b} \map F {c - a, c - b; c; x}$
$\blacksquare$
Also see
- Euler's Integral Representation of Hypergeometric Function
- Kummer's Quadratic Transformation
- Pfaff's Transformation
Source of Name
This entry was named for Leonhard Paul Euler.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 31$: Hypergeometric Functions: Miscellaneous Properties: $31.17$
- 1999: George E. Andrews, Richard Askey and Ranjan Roy: Special Functions: Chapter $\text {2}$. The Hypergeometric Functions
- Weisstein, Eric W. "Euler's Transformation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulersHypergeometricTransformations.html