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A Note on the Asymptotic Expansion of a Ratio of Gamma Functions

Published online by Cambridge University Press:  20 January 2009

Jerry L. Fields
Jerry L. Fields
Affiliation:
Midwest Research Institute, Kansas City, Missouri
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Many problems in mathematical analysis require a knowledge of the asymptotic behaviour of Γ(z + α)/Γ(z + β) for large values of |z|, where α and β are bounded quantities. Tricomi and Erdélyi in (1), gave the asymptotic expansion

where the are the generalised Bernoulli polynomials, see (2), defined by

In this note, we show that if, instead of considering z to be the large variable, we consider a related large variable, (1) can be improved from a computational viewpoint.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 15 , Issue 1 , June 1966 , pp. 43 - 45
Copyright
Copyright © Edinburgh Mathematical Society 1966

References

REFERENCES

(1)Tricomi, F. G. and Erdélyi, A., The asymptotic expansion of a ratio of gamma functions, Pacific J. Math. 1 (1951), 133–142.CrossRefGoogle Scholar
(2)Nörlund, N. E.Vorlesungen über Differenzenrechnung (Springer, Berlin, 1924).CrossRefGoogle Scholar
(3)Watson, G. N.Treatise on the Theory ofBessel Functions, p. 236 (Cambridge University Press, 1958).Google Scholar
(4)Frame, J. S.An approximation to the quotient of gamma functions,Amer. Math. Monthly, 56 (1949), 529–535.Google Scholar
(5)Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher Transcendental Functions, Vol. 1 (McGraw-Hill, 1953).Google Scholar