Geometrically convergent sequences of upper and lower bounds on the Wallis ratio and related expressions

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© Element. Sequences of algebraic upper and lower bounds on the Wallis ratio Γ(x + 1)/Γ(x + 1/2) are given with the relative errors that converge to 0 geometrically and uniformly on any interval of the form [x0, ∞) for x0 > - 1/2 ; moreover, the relative and absolute errors converge to 0 as x→ ∞. These conclusions are based on corresponding results for the digamma function ψ := Γ′ Γ/. Relations with other relevant results are discussed, as well as the corresponding computational aspects. This work was motivated by studies of exact bounds involving the Student probability distribution.

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Mathematical Inequalities and Applications