## Asymptotic expansions and inequalities for hypergeometric functions.(English)Zbl 0897.33001

It was proved by G. D. Anderson, R. W. Barnard, K. C. Richards, M. K. Vamanamurthy and M. Vuorinen [Trans. Am. Math. Soc. 347, No. 5, 1713-1723 (1995; Zbl 0826.33003)] that for $$a,b\in (0,\infty)$$ the function $$g(x)\equiv B(a,b){_2F_1}(a, b;a+ b;x)+ \log(1- x)$$ is strictly increasing from $$(0,1)$$ onto $$(R(a,b),B(a,b))$$, where $$R(a,b)=- 2\gamma- \psi(a)- \psi(b)$$. Here $$\gamma$$, and $$B(a,b)$$ denote the Euler-Mascheroni constant and the beta function, respectively. This result refines Ramanujan’s formula for the asymptotic behavior of the zero-balanced hypergeometric function $${_2F_1}$$.
In the present paper, the authors study the non-zerobalanced case $$a+ b>c$$, with the purpose of seeking some counterparts and generalizations of the abvoe result in this case. For instance, the following result is proved. Theorem. For $$a,b\in(0,\infty)$$, $$a>b$$, and $$c<a+ b$$, define $f(x)\equiv {F(a,b;c,x)\over(1- x)^{c- a-b}},\quad x\in (0,1).$ (i) If $$c>a$$ or $$c< b$$, the function $$f$$ is strictly increasing with range $$(1,D)$$. (ii) If $$b< c<a$$, $$f$$ is strictly decreasing with range $$(D,1)$$. Here $$D$$ is defined by $$D= D(a,b,c)= B(c,a+ b-c)/B(a, b)$$.

### MSC:

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

### Keywords:

Euler-Mascheroni constant; beta function

Zbl 0826.33003
Full Text:

### References:

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