Dixon's Hypergeometric Theorem/Proof 2

Theorem

$\ds \map { {}_3 \operatorname F_2} { { {n, -x, -y} \atop {x + n + 1, y + n + 1} } \, \middle \vert \, 1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {\dfrac n 2 + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {y + \dfrac n 2 + 1} } $

Proof

From Gauss's Hypergeometric Theorem, we have:

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

Therefore:

\(\ds \map F {b + n, c + n; 1 + a + 2 n; 1}\)

\(=\)

\(\ds \dfrac {\map \Gamma {1 + a + 2 n} \map \Gamma {\paren {1 + a + 2 n} - \paren {b + n} - \paren {c + n} } } {\map \Gamma {\paren {1 + a + 2 n} - \paren {b + n} } \map \Gamma {\paren {1 + a + 2 n} - \paren {c + n} } }\)

Gauss's Hypergeometric Theorem

\(\text {(1)}: \quad\)

\(\ds \)

\(=\)

\(\ds \dfrac {\map \Gamma {1 + a + 2 n} \map \Gamma {- b - c + a + 1} } {\map \Gamma {1 + a + n - b } \map \Gamma {1 + a + n - c } }\)

simplifying

Therefore:

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

\(=\)

\(\ds \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {1 + a - b} \map \Gamma {1 + a - c} } \sum_{n \mathop = 0}^\infty \dfrac {\dfrac {\map \Gamma {a + n} } {\map \Gamma a} \dfrac {\map \Gamma {b + n} } {\map \Gamma b} \dfrac {\map \Gamma {c + n} } {\map \Gamma c} } {\dfrac {\map \Gamma {1 + a - b + n} } {\map \Gamma {1 + a - b} } \dfrac {\map \Gamma {1 + a - c + n} } {\map \Gamma {1 + a - c} } } \dfrac {1^n} {n!}\)

Definition of Hypergeometric Function and Rising Factorial as Quotient of Factorials

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {1 + a - b + n} \map \Gamma {1 + a - c + n} } \times \dfrac {\map \Gamma {1 + a + 2 n} } {\map \Gamma {1 + a + 2 n} } \times \dfrac {\map \Gamma {- b - c + a + 1} } {\map \Gamma {1 + a - b - c} }\)

multiplying by $1$ and $1^n = 1$

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {1 + a + 2 n} \map \Gamma {- b - c + a + 1} } \map F {b + n, c + n; 1 + a + 2 n; 1}\)

from $(1)$ above

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {1 + a + 2 n} \map \Gamma {- b - c + a + 1} } \sum_{m \mathop = 0}^\infty \dfrac {\dfrac {\map \Gamma {b + n + m} } {\map \Gamma {b + n} } \dfrac {\map \Gamma {c + n + m} } {\map \Gamma {c + n} } } {\dfrac {\map \Gamma {1 + a + 2 n + m} } {\map \Gamma {1 + a + 2 n} } } \dfrac {1^m} {m!}\)

Definition of Hypergeometric Function and Rising Factorial as Quotient of Factorials

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \sum_{m \mathop = 0}^\infty \dfrac {\map \Gamma {a + n} \map \Gamma {b + n + m} \map \Gamma {c + n + m} } {n! m! \map \Gamma {1 + a + 2 n + m} \map \Gamma {- b - c + a + 1} }\)

simplifying

\(\ds \)

\(=\)

\(\ds \sum_{p \mathop = 0}^\infty \sum_{n \mathop = 0}^p \dfrac {\map \Gamma {a + n} \map \Gamma {b + p} \map \Gamma {c + p} } {n! \paren {p - n}! \map \Gamma {1 + a + n + p} \map \Gamma {- b - c + a + 1} }\)

Letting $p = m + n$, so $m = p - n$ and Product of Absolutely Convergent Series

\(\ds \)

\(=\)

\(\ds \sum_{p \mathop = 0}^\infty \sum_{n \mathop = 0}^p \dfrac {\map \Gamma {a + n} \map \Gamma {b + p} \map \Gamma {c + p} } {n! \paren {p - n}! \map \Gamma {1 + a + n + p} \map \Gamma {- b - c + a + 1} } \times \dfrac {\map \Gamma {1 + a + p} } {\map \Gamma {1 + a + p} } \times \dfrac {\map \Gamma a } {\map \Gamma a } \times \dfrac {p!} {p!} \times \paren {-1}^n \times \paren {-1}^n\)

multiplying by $1$

\(\ds \)

\(=\)

\(\ds \sum_{p \mathop = 0}^\infty \dfrac {\map \Gamma a \map \Gamma {b + p} \map \Gamma {c + p} } {p! \map \Gamma {1 + a + p} \map \Gamma {- b - c + a + 1} } \sum_{n \mathop = 0}^p \dfrac {\paren {\dfrac {\map \Gamma {a + n} } {\map \Gamma a} } \paren {\dfrac {\paren {-1}^n p!} {\paren {p - n}!} } } {\dfrac {\map \Gamma {1 + a + n + p} } {\map \Gamma {1 + a + p} } } \dfrac {\paren {-1}^n} {n!}\)

rearranging terms

\(\ds \)

\(=\)

\(\ds \sum_{p \mathop = 0}^\infty \dfrac {\map \Gamma a \map \Gamma {b + p} \map \Gamma {c + p} } {p! \map \Gamma {1 + a + p} \map \Gamma {- b - c + a + 1} } \map F {a, -p; 1 + a + p; -1}\)

Definition of Hypergeometric Function, Rising Factorial in terms of Falling Factorial of Negative and Rising Factorial as Quotient of Factorials

\(\ds \)

\(=\)

\(\ds \sum_{p \mathop = 0}^\infty \dfrac {\map \Gamma a \map \Gamma {b + p} \map \Gamma {c + p} } {p! \map \Gamma {1 + a + p} \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {p + a + 1} \map \Gamma {\dfrac a 2 + 1} } {\map \Gamma {p + \dfrac a 2 + 1} \map \Gamma {a + 1} }\)

Kummer's Hypergeometric Theorem

\(\ds \)

\(=\)

\(\ds \sum_{p \mathop = 0}^\infty \dfrac {\map \Gamma a \map \Gamma {b + p} \map \Gamma {c + p} } {p! \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} } {\map \Gamma {p + \dfrac a 2 + 1} \map \Gamma {a + 1} }\)

canceling $\map \Gamma {1 + a + p}$

\(\ds \)

\(=\)

\(\ds \sum_{p \mathop = 0}^\infty \dfrac {\map \Gamma a \map \Gamma {b + p} \map \Gamma {c + p} } {p! \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} } {\map \Gamma {p + \dfrac a 2 + 1} \map \Gamma {a + 1} } \times \dfrac {\map \Gamma b} {\map \Gamma b} \times \dfrac {\map \Gamma c} {\map \Gamma c}\)

multiplying by $1$

\(\ds \)

\(=\)

\(\ds \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \sum_{p \mathop = 0}^\infty \dfrac {\dfrac {\map \Gamma {b + p} } {\map \Gamma b } \dfrac {\map \Gamma {c + p} } {\map \Gamma c } } {\dfrac {\map \Gamma {p + \dfrac a 2 + 1} } {\map \Gamma {\dfrac a 2 + 1} } } \dfrac 1 {p!}\)

\(\ds \)

\(=\)

\(\ds \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \map F {b, c; \dfrac a 2 + 1; 1}\)

Definition of Hypergeometric Function and Rising Factorial as Quotient of Factorials

\(\ds \)

\(=\)

\(\ds \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} \map \Gamma {- b - c + \dfrac a 2 + 1} } {\map \Gamma {- b+ \dfrac a 2 + 1} \map \Gamma {- c + \dfrac a 2 + 1} }\)

Gauss's Hypergeometric Theorem

We now have:

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

\(=\)

\(\ds \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} \map \Gamma {- b - c + \dfrac a 2 + 1} } {\map \Gamma {- b+ \dfrac a 2 + 1} \map \Gamma {- c + \dfrac a 2 + 1} }\)

\(\ds \leadsto \ \ \)

\(\ds \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {- b + a + 1, - c + a + 1} } \, \middle \vert \, 1}\)

\(=\)

\(\ds \dfrac {\map \Gamma {- b + a + 1} \map \Gamma {- c + a + 1} } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} \map \Gamma {- b - c + \dfrac a 2 + 1} } {\map \Gamma {- b+ \dfrac a 2 + 1} \map \Gamma {- c + \dfrac a 2 + 1} }\)

Substituting $a = n$, $b = -x$ and $c = -y$, we obtain:

\(\ds \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {1 + a - b, 1 + a - c} } \, \middle \vert \, 1}\)

\(=\)

\(\ds \dfrac {\map \Gamma {- b + a + 1} \map \Gamma {- c + a + 1} } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} \map \Gamma {- b - c + \dfrac a 2 + 1} } {\map \Gamma {- b + \dfrac a 2 + 1} \map \Gamma {-c + \dfrac a 2 + 1} }\)

before substitution

\(\ds \leadsto \ \ \)

\(\ds \map { {}_3 \operatorname F_2} { { {n, -x, -y} \atop {x + n + 1, y + n + 1} } \, \middle \vert \, 1}\)

\(=\)

\(\ds \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {\dfrac n 2 + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {y + \dfrac n 2 + 1} }\)

after substitution

$\blacksquare$

Source of Name

This entry was named for Alfred Cardew Dixon.

Sources

• 1935: W.N. Bailey: Generalized Hypergeometric Series Chapter $\text {3}$. The hypergeometric series