Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An affirmative answer to an open problem on Ramanujan's asymptotic formula of zero-balanced hypergeometric function

Miao-kun Wang, Zhen-hang Yang, Tie-hong Zhao

TL;DR

The paper addresses Open Problem AVV (1997) by proving that a function $Q_p(x)$, built from Ramanujan's asymptotic formula for the zero-balanced hypergeometric function and normalized via $d=e^{R(a,b)}$, is absolutely monotone on $(0,1)$ for all $a,b>0$ with $a+b\le1$ and $2\le p\le R(a,b)$. The authors derive a recurrence for the Maclaurin coefficients $\alpha_n$ of $Q_p(x)$ and reduce the positivity of these coefficients to the intersection of four index sets $E_1,E_2,E_3,E_4$, showing it equals $[2,R(a,b)]$; this yields $\alpha_n\ge0$ for all $n$, hence absolute monotonicity. Taking $p=R(a,b)$ provides a direct affirmative answer to AVV, thus refining known monotonicity results for generalized elliptic and hypergeometric constructs. The work also develops Anderson–Vamanamurthy–Vuorinen style beta and Ramanujan $R$-function properties, which may yield sharper inequalities and expansions for related special functions.

Abstract

In this paper, by using recurrence method and new properties of beta function and Ramanujan $R$-function, we prove the absolute monotonicity result for a function related to Ramanujan asymptotic formula of zero-balanced hypergeometric function, which gives an affirmative answer to an open problem proposed by Anderson Vamanamurthy and Vuorinen \cite{Anderson-CIIQM-JW-1997} in 1997.

An affirmative answer to an open problem on Ramanujan's asymptotic formula of zero-balanced hypergeometric function

TL;DR

The paper addresses Open Problem AVV (1997) by proving that a function , built from Ramanujan's asymptotic formula for the zero-balanced hypergeometric function and normalized via , is absolutely monotone on for all with and . The authors derive a recurrence for the Maclaurin coefficients of and reduce the positivity of these coefficients to the intersection of four index sets , showing it equals ; this yields for all , hence absolute monotonicity. Taking provides a direct affirmative answer to AVV, thus refining known monotonicity results for generalized elliptic and hypergeometric constructs. The work also develops Anderson–Vamanamurthy–Vuorinen style beta and Ramanujan -function properties, which may yield sharper inequalities and expansions for related special functions.

Abstract

In this paper, by using recurrence method and new properties of beta function and Ramanujan -function, we prove the absolute monotonicity result for a function related to Ramanujan asymptotic formula of zero-balanced hypergeometric function, which gives an affirmative answer to an open problem proposed by Anderson Vamanamurthy and Vuorinen \cite{Anderson-CIIQM-JW-1997} in 1997.

Paper Structure

This paper contains 8 sections, 12 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['MT']}
  3. Substep 2.1: $E_{3}\cap E_{4}=\left\{ p:2\leq p\leq R\left( a,b\right) \right\}$
  4. Substep 2.2: $E_{3}\cap E_{4}\subset E_{1}$
  5. Substep 2.3: $E_{3}\cap E_{4}\subseteq E_{2}$
  6. Proof of Theorem \ref{['MT']}
  7. Some properties of Beta function and Ramanujan $R$-function
  8. Concluding remarks

Key Result

Theorem 1

Let $p>0$ and $a,b\in (0,1)$ with $a+b\leq 1$. If $2\leq p\leq R\left( a,b\right)$, then the function $x\mapsto Q_{p}(x)$ defined by (Qp) is absolutely monotonic on $\left( 0,1\right)$. In other words, $Q_{p}(x)$ has a Maclaurin expansion $\sum_{n=0}^{\infty }\alpha _{n}x^{n}$ with non-negative coef

Theorems & Definitions (29)

  • Conjecture 1
  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Proposition 1: Substep 2.1
  • proof
  • Proposition 2: Substep 2.2
  • proof
  • Lemma 1
  • ...and 19 more