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

Ponnusamy, S.; Vuorinen, M.

Mathematika 44, No. 2, 278-301 (1997).

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 \(D= D(a,b,c)= B(c,a+ b-c)/B(a, b)\).

Reviewer: M.Vuorinen (Helsinki)

MSC:

33C05

Classical hypergeometric functions, \({}_2F_1\)

Keywords:

Euler-Mascheroni constant; beta function

Citations:

Zbl 0826.33003

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