Generalization of any product similar to the Gamma function

Question

So, the Gamma function generalizes the following product: $$n!=1\cdot 2\cdot 3\cdot..........n$$ Because it is only defined when $n$ is a natural number. But I thought about the following product: $$n{$}=1\cdot (1+d)\cdot (1+2d) \cdot..................n$$ Clearly this product is also defined only when $n$ is of the form $1+kd$ where $k$ is a natural number. Can this product also be generalized when $n$ is not of that form? For example, the product: $$f(n)=1\cdot 1.3\cdot 1.6\cdot 1.9\cdot 2.2......n$$ Clearly, this product is only defined when $n$ is of the form $1+0.3k$. Can this product be generalized?

Answer 1

As Chappers commented$$P_n=\prod_{i=0}^n(1+id)=d^n \left(1+\frac{1}{d}\right)_n$$ where appears Pochhammer symbol.

This can also be expressed in terms of the gamma function $$P_n=d^n\frac{ \Gamma \left(n+1+\frac{1}{d}\right)}{\Gamma \left(1+\frac{1}{d}\right)}$$