For $a,b>0,m\in \{1,2,3,...\}$ $$ \frac{a*m \pi}{a+b}\;{}_2F_3\left( \frac{a+1}{2},\frac{a+2}{2}; \frac{3}{2},\frac{a+b+1}{2},\frac{a+b+2}{2}; \frac{-m^2\pi^2}{4} \right) \\ =\frac{-i}{2} ( {}_1F_1(a;a+b;i m \pi)-{}_1F_1(a;a+b;-i m \pi)) \\ =\frac{1}{\mathbf{B}(a,b)}\int_{0}^{1} u^{a-1} (1-u)^{b-1} \sin (m\pi u) du $$
1) Is there a simpler proof of this relation than the one here? $I_k=\int_0^1 \frac{1}{\mathbf{B}(\alpha , \beta )} \cos^k (\pi \theta) \theta^{\alpha -1} (1-\theta)^{\beta -1}d\theta $ .
2) For what domain of $m$ it is true? is it work for not integer values?
A Hypergeometric Puzzle is another question that extracted from $I_k=\int_0^1 \frac{1}{\mathbf{B}(\alpha , \beta )} \cos^k (\pi \theta) \theta^{\alpha -1} (1-\theta)^{\beta -1}d\theta $
R code
a<<-4
b<<-3
m<<-5
#
fn<-function(u){
ret<-sin(m*pi*u)*dbeta(u,a,b)
return(ret)
}
## 1F1
U<-c(a)
L<-c(a+b)
z<-(m*pi)*1i
-((1i)/2)*(genhypergeo(U,L,z) -genhypergeo(U,L,-z))
integrate(fn,lower=0,upper=1,abs.tol=1.7e-10)
#output
#> -((1i)/2)*(genhypergeo(U,L,z) -genhypergeo(U,L,-z))
#[1] -0.02945569+0i
#> integrate(fn,lower=0,upper=1,abs.tol=1.7e-10)
#-0.02945569 with absolute error < 1.9e-12
#2F3
U2<-c((a+1)/2,(a+2)/2)
L2<-c(3/2,(a+b+1)/2,(a+b+2)/2)
z2<--(m^2)*(pi^2)/4
((a*m*pi)/(a+b))*genhypergeo(U2,L2,z2)
#> ((a*m*pi)/(a+b))*genhypergeo(U2,L2,z2)
#[1] -0.02945569