Is there a relationship between $a_i$ and $a_{i+1}$, where $a_i = {}_2{F_1}( - \delta ,i + 1;1 - \delta ; - z )$.

Question

Define $a_i = {}_2{F_1}( - \delta ,i + 1;1 - \delta ; - z )$, $i=0,1,2,\cdots,$ $0< \delta<1$, $z\ge 0$. So, what is the relationship between $a_i$ and $a_{i+1}$?

Answer 1

One can use the simplest of Gauss' contiguous relations for $_2F_1$: $$(b-a)\ _2F_1(a,b;c;z)=b\ _2F_1(a,b+1;c;z)-a\ _2F_1(a+1,b;c;z),$$ and recall that $_2F_1(a,b;a;z)=(1-z)^{-b}$. Thus, $a\gets-\delta$, $b\gets i+1$, $c\gets 1-\delta$, $z\gets-z$ yields $$\bbox[5px,border:3px solid]{(i+1)a_{i+1}=(i+1+\delta)a_i-\delta(1+z)^{-i-1}.}$$