In Euler's "Differential calculus" we can find gamma function, as special case of interpolation of production, which factors formed by arithmetic progression: $$f(a,b,w)=a^w\prod\limits_{n=0}^{\infty}\frac{(a+nb)^{1-w}(a+(n+1)+b)^w}{a+b(n+w)}$$ So we can say, that $$f(1,1,w)=\Gamma(1+w)$$ From that he find: $$f(1,1,\frac{1}{2})=\Gamma(\frac{3}{2})=\frac{\sqrt{\pi}}{2}$$ $$f(1,2,\frac{1}{2})=\sqrt{\frac{2}{\pi}}$$ Last result can be expressed as interpolation for odd double factorial. Also using this function we can have any $n$-th factorials.
Can we call this function as generalization for $\Gamma(z)$? Is there any very well known function, which have the same character?
If I made some mistakes, sorry for my English.