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Derivation of the Legendre Duplication Formula 📂Functions

Derivation of the Legendre Duplication Formula

Formula

Γ(2r)=22r−1πΓ(r)Γ(12+r) \Gamma (2r) = {{2^{ 2r - 1} } \over { \sqrt{ \pi } } } \Gamma \left( r \right) \Gamma \left( {{1} \over {2}} + r \right)

Description

While the splitting shape may not be pretty, the fact that factors can be divided into smaller ones is certainly useful. The derivation itself is not very difficult if one uses an auxiliary lemma derived from the beta function.

Derivation

For B(p,q)=Γ(p)Γ(q)Γ(p+q)=∫01tp−1(1−t)q−1dt B(p,q) = {{\Gamma (p) \Gamma (q)} \over {\Gamma (p+q) }} = \int_{0}^{1} t^{p-1} (1-t)^{q-1} dt , if it is said that r:=p=qr:= p=q Γ(r)Γ(r)Γ(2r)=∫01tr−1(1−t)r−1dt {{\Gamma (r) \Gamma (r)} \over {\Gamma (2r) }} = \int_{0}^{1} t^{r-1} (1-t)^{r-1} dt When substituting t=1+s2\displaystyle t = {{1+s} \over {2}} for λ(s):=(1−s2)r−1\lambda (s) := \left( 1 - s^2 \right)^{r-1}, since it is an even function, Γ(r)Γ(r)Γ(2r)=12∫−11(1+s2)r−1(1−s2)r−1ds=121+2(r−1)∫−11(1−s2)r−1ds=21−2r⋅2∫01(1−s2)r−1ds \begin{align*} {{\Gamma (r) \Gamma (r)} \over {\Gamma (2r) }} =& {{1} \over {2}} \int_{-1}^{1} \left( {{1+s} \over {2}} \right)^{r-1} \left( {{1-s} \over {2}} \right)^{r-1} ds \\ =& {{1} \over {2^{1 + 2(r-1)} }} \int_{-1}^{1} \left( 1 - s^2 \right)^{r-1} ds \\ =& 2^{1 - 2r} \cdot 2 \int_{0}^{1} \left( 1 - s^2 \right)^{r-1} ds \end{align*}

Corollary of the trigonometric expression of the beta function: B(x,y)=2∫01t2x−1(1−t2)y−1dt B(x,y) = 2 \int_{0}^{1} t^{2x-1} \left( 1 - t^2 \right)^{y-1} dt

By substituting x=12\displaystyle x = {{1} \over {2}} and y=ry = r into the above formula, B(12,r)=2∫01(1−t2)r−1dt B \left( {{1} \over {2}} , r \right) = 2 \int_{0}^{1} \left( 1 - t^2 \right)^{r-1} dt Therefore, Γ(r)Γ(r)Γ(2r)=21−2rB(12,r)=21−2rΓ(12)Γ(r)Γ(12+r) {{\Gamma (r) \Gamma (r)} \over {\Gamma (2r) }} = 2^{1 - 2r} B \left( {{1} \over {2}} , r \right) = 2^{1 - 2r} {{\Gamma \left( {{1} \over {2}} \right) \Gamma (r)} \over {\Gamma \left( {{1} \over {2}} + r \right) }} is obtained. Since from the reflection formula Γ(12)=π\displaystyle \Gamma \left( {1 \over 2} \right) = \sqrt{\pi} Γ(r)Γ(2r)=21−2rπΓ(12+r) {{\Gamma (r)} \over {\Gamma (2r) }} = 2^{1 - 2r} {{\sqrt{\pi} } \over {\Gamma \left( {{1} \over {2}} + r \right) }} Organizing with respect to Γ(2r)\Gamma (2r) yields Γ(2r)=22r−1πΓ(r)Γ(12+r) \Gamma (2r) = {{2^{2r-1} } \over { \sqrt{ \pi } } } \Gamma \left( r \right) \Gamma \left( {{1} \over {2}} + r \right)

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