Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$. An application $h$ from a non-trivial interval $J$ of $\mathbf{R}$ to $\mathbf{R}$ is said to be log-convex if, and only if, it takes values in $\mathbf{R}_+^*$ and $\ln \circ h$ is convex on $J$. The functional equation referred to as (1) is: $$\forall x \in I, (x+1)f(x) = (x+2)f(x+2)$$ Conclude that $f$ is the unique application from $I$ to $\mathbf{R}$, which is log-convex, which satisfies (1) and such that $$f(0) = \frac{\pi}{2}$$
Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$. An application $h$ from a non-trivial interval $J$ of $\mathbf{R}$ to $\mathbf{R}$ is said to be log-convex if, and only if, it takes values in $\mathbf{R}_+^*$ and $\ln \circ h$ is convex on $J$. The functional equation referred to as (1) is:
$$\forall x \in I, (x+1)f(x) = (x+2)f(x+2)$$
Conclude that $f$ is the unique application from $I$ to $\mathbf{R}$, which is log-convex, which satisfies (1) and such that
$$f(0) = \frac{\pi}{2}$$