## Ramanujan’s cubic transformation inequalities for zero-balanced hypergeometric functions.(English)Zbl 1350.33007

The hypergeometric function is defined for $$|x| <1$$ by the power series $F(a, b; c; x)={}_{2}F_{1}(a,b;c;x)=\sum _{n=0}^{\infty }{\frac {(a,n)(b,n)}{(c,n)}}{\frac {x^{n}}{n!}}.$ It is defined if $$c$$ is different from non-positive integers. With the symbol $$(a,n)$$ we denote the shifted factorial function: $$(a,n)=1$$ if $$n=0$$ and $$(a,n)=a(a+1)\cdots (a+n-1)$$ if $$n>0$$, for $$a\neq0$$. Also, $$F(a, b; c; x)$$ is called zero-balanced if $$c=a+b$$.
This paper focus on Ramanujan’s cubic transformation, which is defined as $F\left(\frac{1}{3}, \frac{2}{3}; 1; 1-\left(\frac{1-r}{1+2r}\right)^3\right)=(1+2r) F\left( \frac{1}{3}, \frac{2}{3}; 1; r^3\right),$
$F\left(\frac{1}{3}, \frac{2}{3}; 1; \left(\frac{1-r}{1+2r}\right)^3\right)=\frac{1+2r}{3} F\left( \frac{1}{3}, \frac{2}{3}; 1; 1-r^3\right).$ The authors prove a generalization of Ramanujan’s cubic transformation, in the form of an inequality, as shown in Section 2 (main results).

### MSC:

 33C05 Classical hypergeometric functions, $${}_2F_1$$

DLMF
Full Text:

### References:

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