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Simple, accurate, asymptotic estimates for the ratio of two Gamma functions

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Several new, accurate, asymptotic estimates for the ratio of two Gamma functions, \(Q_{\varGamma }(x,a,b):=\frac{\varGamma (x+a)}{\varGamma (x+b)}\), are obtained on the basis of Stirling’s approximation formula for the \(\varGamma \) function.

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Notes

  1. especially for \(a,b\in (0,1)\) and \(x>0\)

  2. Consider that \(\sum _{i=1}^0 x_i=0\), by definition.

  3. Consider Footnote 2.

  4. All graphics and calculations in this paper are made using the Mathematica [23] computer system.

  5. Using for \(u:=\tfrac{b}{x}\), \(v:=\tfrac{|a-b|}{x+b}\) and \(w:=2\tfrac{7}{12x}\), the relations \(u,v\le \tfrac{1}{3}\), \(uv\le \tfrac{1}{9}\) and \(uv+uw+vw+uvw\le \tfrac{1}{3}v+\tfrac{1}{3}w+\tfrac{1}{3}w+\tfrac{1}{9}w \).

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  1. University of Ljubljana, 1 Ljubljana, 386, Slovenia

    Vito Lampret

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Lampret, V. Simple, accurate, asymptotic estimates for the ratio of two Gamma functions. RACSAM 115, 40 (2021). https://doi.org/10.1007/s13398-020-00962-9

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