Given a function $F(z)$, the function $F(z-m)$ is called the Taylor shift of $F(z)$. If $F(z)$ is a (general) hypergeometric function, is $F(z-m)$ some simple transformation of a hypergeometric function. Typical transforms of $F(z)$ are $A(z) F(B(z))$ where $A(z), B(z)$ are polynomials.

More precisely, suppose $F(z)={}_pF_q((a);(b);z)$ where $(a),(b)$ are the parameters, is $F(z-m)$ is some simple transformation of a hypergeometric function.

For example, $F(z)={}_1F_0(a;;z)$, I can show that $$ F(z-m) = (1+m)^{-a} F\big(\frac{z}{1+m}\big) $$ Taylor shift is very important for efficient algorithms. Also, any pointers or references would be appreciated. This is my first question in mathoverflow. Not sure how to get latex to be processed.