From the fact that $1\times3\times5\times\ldots\times(2n-1)=\frac{(2n)!}{2^nn!}$, we can show that $$ (1-x)^{1/2}=\sum_{n=0}^\infty \frac{(2n-2)!}{(n-1)!n!2^{2n+1}}x^n. $$ However, can I do the same thing for $(1-x)^{1/4}$? I need to find a closed form of $3\times 7\times11\times\ldots\times(4n-1)$ in terms of factorial, similar to the identity above. I tried wolfram, but it only gives it in terms of gamma function without proof.
How can I express the product $$ \prod_{r=0}^n ({ar+d}) $$ in a closed form of factorial or gamma function?
$$P_n=\prod_{r=0}^n ({ar+d})=d a^n \left(\frac{a+d}{a}\right)_n$$ using Pochhammmer symbols.
In terms of the gamma function $$P_n=a^{n+1}\frac{\Gamma \left(n+1+\frac{d}{a}\right)}{\Gamma \left(\frac{d}{a}\right)}$$