On Gautschi & Stirling Identities, Asymptotics and Inequalities for the Pi (or Gamma) Function

TL;DR

The article derives integral-identity based two-sided bounds for Stirling-type asymptotics of piecewise logarithmic interpolations of the pi/gamma function, connecting Gautschi's inequality with Stirling-type bounds via $\widehat{\Pi}$ and $x\widehat{!}$. By expressing mismatches $m_d$ and $\widehat{m}_d$ as integrals, it obtains general and improved bounds that are asymptotically optimal, including Burnside-type refinements for the Burnside variant. The results yield explicit, easy-to-check inequalities for $\Pi(x)/S_d(x)$ and $\widehat{\Pi}(x,\alpha)/S_d(x+\alpha)$, with concrete constants for $d=0,1$ and $d=1/2$, and are complemented by numerical comparisons. The methods are elementary and unify several known bounds while suggesting directions for further tightening, including conjectured bounds independent of $x$ under certain conditions. These bounds have potential applications in the analysis of Gevrey-smooth functions and in numerical approximations of factorials and gamma-related quantities.

Abstract

We derive two-sided bounds for a class of Stirling-type asymptotic formulas for piecewise logarithmic interpolations of the pi function, and hence also for the factorials and the gamma functions. The bounds are derived by first proving some integral identity versions of Gautschi's inequality and a class of Stirling-type asymptotic formulas, and then bounding these integrals by asymptotically optimal bounds. Additionally, all the proofs given rely only on common elementary arguments and connect, generalise and possibly improve various results that have been published previously. Lastly, we provide numerical comparisons concerning the effectiveness and behaviour of the bounds and approximations in a graphical manner, which clearly indicate that the bounds are asymptotically optimal.

Figures (9)

• Figure 1: Visualisation of the upper bounds for $\widehat{\iota}(x, \alpha)$ in terms of $x+\alpha$.
• Figure 2: Visualisation of the two-sided bounds for $m_{1/2}(x)$ in terms of $x$.
• Figure 3: Visualisation of the two-sided bounds for $m_{0}(x)$ in terms of $x$.
• Figure 4: Visualisation of the two-sided bounds for $m_{1}(x)$ in terms of $x$.
• Figure 5: Visualisation of the two-sided bounds for $m_{-1/2}(x)$ in terms of $x$.

Theorems & Definitions (37)

• Lemma 1
• Corollary 2
• Corollary 3
• Lemma 4
• Lemma 5
• Theorem 6
• Lemma 7
• Lemma 8
• Lemma 9
• Corollary 10