ASYMPTOTIC EXPANSIONS FOR RATIOS OF PRODUCTS OF GAMMA FUNCTIONS

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ASYMPTOTIC EXPANSIONS FOR RATIOS OF PRODUCTS OF GAMMA FUNCTIONS

WOLFGANG BÜHRING Received 6 May 2002

An asymptotic expansion for a ratio of products of gamma functions is derived.

2000 Mathematics Subject Classification: 33B15, 33C20.

1. Introduction. An asymptotic expansion for a ratio of products of gamma functions has recently been found [2], which, with

s1=b1−a1−a2, (1.1)

may be written as Γ

a1+n Γ

a2+n Γb1+n

Γ

−s1+n=1+ M m=1

s1+a1

m

s1+a2

m

(1)_m

1+s1−n

m +O n^−M−1

(1.2)

asn→ ∞. Here, we make use of the Pochhammer symbol

(x)n=x(x+1)···(x+n−1)=Γ(x+n)/Γ(x). (1.3)

The special case when b1=1 of formula (1.2) had been stated earlier by Dingle [3], and there were proofs by Paris [8] and Olver [6,7].

The proof of (1.2) is based on the formula for the analytic continuation near unit argument of the Gaussian hypergeometric function. For the more general hypergeometric functions

p+1Fp



a1,a2,...,a_p+1 b1,...,bp

z



= ∞ n=0

a1

n

a2

n···

a_p+1 n

b1

n···

bp

n(1)n zⁿ,

|z|<1 , (1.4)

the analytic continuation nearz=1 is known too, and this raises the question as to whether a sufficiently simple asymptotic expansion can be derived in a similar way for a ratio of products of more gamma function factors. This is indeed the case, and it is the purpose of this work to present such an expansion.

2. Derivation of the asymptotic expansion. The analytic continuation of the hypergeometric function near unit argument may be written as (see [1])

Γa1 Γa2

···Γa_p+1 Γb1

···Γb_{p p+}1F_p



a1,a2,...,a_p+1

b1,...,b_p z





= ^∞

m=0

g_m(0)(1−z)^m+(1−z)^{sp ∞}

m=0

g_ms_p

(1−z)^m,

(2.1)

where

s_p=b1+···+b_p−a1−a2−···−a_p+1 (2.2) and the coefficientsgm are known. While thegm(0) are not needed for the present purpose, theg_m(s_p)are [1]

g_ms_p

=(−1)^m

a1+sp

m

a2+sp

mΓ

−sp−m (1)m

× m k=0

(−m)k

a1+s_p

ka2+s_p

k

A^(p)_k ,

(2.3)

where the coefficientsA^(p)_k are to be shown below.

The left-hand sideLof (2.1) is L=Γ

a1+n Γ

a2+n

···Γ

a_p+1+n Γ

b1+n

···Γ bp+n

Γ(1+n) zⁿ. (2.4) The asymptotic behaviour, asn→ ∞, of the coefficients of this power series is governed [4,5,10] by the termsRon the right-hand side which, atz=1, are singular

R= ^∞

m=0

g_ms_p

(1−z)^sp+m. (2.5)

Expanded by means of the binomial theorem in its hypergeometric-series form, this is

R= ∞ m=0

gm sp^∞

n=0

−s_p−m

n

Γ(1+n) zⁿ. (2.6)

Interchanging the order of summation (and making use of the reflection for- mula of the gamma function), we may get

R=^∞

n=0

Γ

−sp+n Γ(1+n)

∞ m=0

(−1)^mg_ms_p 1 Γ

−sp−m

1+sp−n

m

zⁿ. (2.7)

ASYMPTOTIC EXPANSIONS FOR RATIOS OF PRODUCTS 1169 Comparison of the coefficients of the two power series for R and L, which should agree asymptotically asn→ ∞, then leads to

Γa1+n

Γa2+n

···Γa_p+1+n Γb1+n

···Γb_p+n Γ

−s_p+n

∼ ∞ m=0

(−1)^mgm

sp 1

Γ

−s_p−m

1+s_p−n

m

.

(2.8)

Insertingg_mfrom (2.3) and keeping the first M+1 terms of the asymptotic series, we get the following theorem.

Theorem2.1. Withs_p=b1+ ··· +b_p−a1−a2− ··· −a_p+1, we have the asymptotic expansion

Γa1+n

Γa2+n

···Γa_p+1+n Γb1+n

···Γb_p+n Γ

−s_p+n

=1+ M m=1

a1+s_p

m

a2+s_p

m

(1)_m

1+s_p−n

m

m k=0

(−m)_k a1+s_p

k

a2+s_p

k

A^(p)_k +O n^−M−1

(2.9) asn→ ∞.

The simple formula (1.2), corresponding top=1, can be recovered from this theorem if we defineA⁽¹⁾₀ =1,A⁽¹⁾_k =0 fork >0, so that the sum overkis then equal to 1 and disappears. The coefficients for largerp can be found in [1], but a few of them are displayed again here for convenience

A⁽²⁾_k =

b2−a3

k

b1−a3

k

k! ,

A⁽³⁾_k = k k2=0

b3+b2−a4−a3+k2

k−k2

b1−a3

k−k2

b3−a4

k2

b2−a4

k2

k−k2!k2! ,

A⁽⁴⁾_k = k k2=0

b4+b3+b2−a5−a4−a3+k2

k−k2

b1−a3

k−k2

k−k2

!

×

k2

k₃=0

b4+b3−a5−a4+k3

k₂−k3

b2−a4

k₂−k3

k2−k3

!

b4−a5

k₃

b3−a5

k₃

k3! . (2.10) Forp=3,4,...,several other representations are possible [1] such as

A⁽³⁾_k =

b3+b2−a4−a3

k

b1−a3

k

k!

×3F2



 b3−a4,b2−a4,−k b3+b2−a4−a3,1+a3−b1−k

1





(2.11)

or

A⁽³⁾_k =

b1+b3−a3−a4

kb2+b3−a3−a4

k

k!

×3F2



 b3−a3,b3−a4,−k b1+b3−a3−a4,b2+b3−a3−a4

1



.

(2.12)

Forp=2, (2.9) may be simply written as Γa1+n

Γa2+n

Γa3+n Γ

b1+n Γ

b2+n Γ

−s2+n

=1+ M m=1

a1+s2

m

a2+s2

m

(1)_m

1+s2−n

m 3F2



b2−a3,b1−a3,−m a1+s2,a2+s2

1



+O n^−M−1

,

(2.13) wheres2=b1+b2−a1−a2−a3.

3. Additional comments. The derivation of the theorem is based on the continuation formula (2.1) which holds, as it stands, only ifs_pis not equal to an integer. Nevertheless, the theorem is valid without such a restriction. This can be verified if the derivation is repeated starting from any of the continuation formulas for the exceptional cases [1]. Instead of or in addition to the binomial theorem, the expansion

(1−z)^mln(1−z)= ∞ n=1

c_nzⁿ (3.1)

is then needed for integerm≥0, where cn= −1

n(−1)^mΓ(1+m)Γ(n−m)

Γ(n) (3.2)

forn > m, while the coefficients are not needed here forn≤m.

The theorem has been proved here for any sufficiently large positive integer nonly. On the basis of the discussion in [2], it can be suspected that the the- orem may be theoretically valid (although less useful) in the larger domain of the complexn-half-plane Re(sp+a1+a2−1+n)≥0.

Expansions for ratios of even more general products of gamma functions are treated in a recent monograph by Paris and Kaminski [9].

References

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[2] ,An asymptotic expansion for a ratio of products of gamma functions, Int.

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ASYMPTOTIC EXPANSIONS FOR RATIOS OF PRODUCTS 1171 [3] R. B. Dingle,Asymptotic Expansions: Their Derivation and Interpretation, Aca-

demic Press, London, 1973.

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[5] F. W. J. Olver,Asymptotics and Special Functions, Computer Science and Applied Mathematics, Academic Press, New York, 1974.

[6] ,Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations, Methods Appl. Anal.1(1994), no. 1, 1–13.

[7] ,On an asymptotic expansion of a ratio of gamma functions, Proc. Roy.

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[9] R. B. Paris and D. Kaminski,Asymptotics and Mellin-Barnes integrals, Encyclo- pedia of Mathematics and its Applications, vol. 85, Cambridge University Press, Cambridge, 2001.

[10] R. Schäfke and D. Schmidt,The connection problem for general linear ordinary differential equations at two regular singular points with applications in the theory of special functions, SIAM J. Math. Anal.11(1980), no. 5, 848–862.

Wolfgang Bühring: Physikalisches Institut, Universität Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany

E-mail address:[email protected]