On the asymptotic expansion of a ratio of gamma functions

An asymptotic expansion of a ratio of products of gamma functions is derived. It generalizes a formula which was stated by Dingle, first proved by Paris, and recently reconsidered by Olver

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Corrigendum and addendum to ‘asymptotic series for double zeta, double gamma and Hecke L-functions’

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004101005631 ◽

2002 ◽

Vol 132 (2) ◽

pp. 377-384 ◽

Cited By ~ 5

Author(s):

KOHJI MATSUMOTO

Keyword(s):

Asymptotic Expansion ◽

Asymptotic Series ◽

Gamma Functions ◽

Double Zeta ◽

Error Terms

Refined expressions are given for the error terms in the asymptotic expansion formulas for double zeta and double gamma functions, proved in the author's former paper [2]. Some inaccurate claims in [2] are corrected.

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The Asymptotic Expansion of the Incomplete Gamma Functions

SIAM Journal on Mathematical Analysis ◽

10.1137/0510071 ◽

1979 ◽

Vol 10 (4) ◽

pp. 757-766 ◽

Cited By ~ 56

Author(s):

N. M. Temme

Keyword(s):

Asymptotic Expansion ◽

Gamma Functions ◽

Incomplete Gamma Functions

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

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500013171 ◽

1966 ◽

Vol 15 (1) ◽

pp. 43-45 ◽

Cited By ~ 19

Author(s):

Jerry L. Fields

Keyword(s):

Asymptotic Expansion ◽

Asymptotic Behaviour ◽

Mathematical Analysis ◽

Bernoulli Polynomials ◽

Gamma Functions ◽

Image Position

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 expansionwhere the are the generalised Bernoulli polynomials, see (2), defined byIn 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.

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I.—Induction Proofs of the Relations between certain Asymptotic Expansions and Corresponding Generalised Hypergeometric Series.

Proceedings of the Royal Society of Edinburgh ◽

10.1017/s0370164600010993 ◽

1939 ◽

Vol 58 ◽

pp. 1-13 ◽

Cited By ~ 4

Author(s):

T. M. MacRobert

Keyword(s):

Asymptotic Expansion ◽

Asymptotic Expansions ◽

Hypergeometric Functions ◽

Hypergeometric Series ◽

Multiple Integral ◽

Contour Integral ◽

Gamma Functions ◽

Induction Proofs ◽

The Subject

The subject of this paper was studied by Orr (Camb. Phil. Trans., vol. xvii, 1898, pp. 171–199; 1899, pp. 283–290), and later by Barnes (Proc. Lond. Math. Soc., ser. 2, vol. v, 1906, pp. 59–116), in whose paper a number of references to earlier work on the subject are given. Formulæ equivalent to (9), (18), and (20) below were given by, Barnes, who derived them by his well‐known method of integrating products and quotients of Gamma Functions. In this paper the formulæ are deduced by induction from simpler formulæ which are established in section 2. In order to simplify the notation, four functions, denoted by P, E, Q, and H, are introduced, the first two in section 3, the last two in section 5. The P and Q functions are merely generalised hypergeometric functions multiplied by convenient factors. The E function, which is equivalent to Barnes's contour integral, is defined as a multiple integral, and from it the asymptotic expansion, with a useful form for the remainder, is easily derived. The H function is a multiple of the E function.

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