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


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


Zbl 0826.33003
Full Text: DOI


[1] Askey, Uspekhi Mat. Nauk 45 pp 33– (1990)
[2] DOI: 10.2307/2154966 · Zbl 0826.33003
[3] DOI: 10.2307/2153171 · Zbl 0802.33001
[4] Abramowitz, Handbook of mathematical functions with formulas. Graphs and mathematical tables (1965) · Zbl 0171.38503
[5] Whittaker, A Course of Modern Analysis (1958)
[6] Varchenko, Proc. Internal. Congr. Math pp 281– (1990)
[7] Berndt, Ramanujan’s Notebooks (1989)
[8] Mitrinovic, Analytic inequalities, Die Grundlehren der math. Wissenschaften pp 165– (1970)
[9] Rainville, Special functions (1960)
[10] Evans, Ramanujan’s second notebook: asymptotic expansions for hypergeometric series and related Junctions, In Ramanujan Revisited: Proc. of the Centena y Conference Univ. of Illinois at Urbana-Champaign pp 537– (1988) · Zbl 0646.33003
[11] DOI: 10.2307/2008180 · Zbl 0607.33002
[12] Biernacki, Ann. Univ. M. Curie-Sktodowska 2 pp 134– (1995)
[13] Ramanujan, The lost notebook and other unpublished papers (1988) · Zbl 0639.01023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.