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).


33C05 Classical hypergeometric functions, \({}_2F_1\)


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