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16 Generalized Hypergeometric Functions and Meijer G-FunctionGeneralized Hypergeometric Functions

§16.11 Asymptotic Expansions

Contents

§16.11(i) Formal Series

For subsequent use we define two formal infinite series, Ep,q(z) and Hp,q(z), as follows:

16.11.1 Ep,q(z)=(2π)(p-q)/2κ-ν-(1/2)κz1/κk=0ck(κz1/κ)ν-k,
p<q+1,
16.11.2 Hp,q(z)=m=1pk=0(-1)kk!Γ(am+k)(=1mpΓ(a-am-k)/=1qΓ(b-am-k))z-am-k.

In (16.11.1)

16.11.3 κ =q-p+1,
ν =a1++ap-b1--bq+12(q-p),

and

16.11.4 c0 =1,
ck =-1kκκm=0k-1cmek,m,
k1,

where

16.11.5 ek,m=j=1q+1(1-ν-κbj+m)κ+k-m(=1p(a-bj)/=1jq+1(b-bj)),

and bq+1=1.

It may be observed that Hp,q(z) represents the sum of the residues of the poles of the integrand in (16.5.1) at s=-aj,-aj-1,, j=1,,p, provided that these poles are all simple, that is, no two of the aj differ by an integer. (If this condition is violated, then the definition of Hp,q(z) has to be modified so that the residues are those associated with the multiple poles. In consequence, logarithmic terms may appear. See (15.8.8) for an example.)

§16.11(ii) Expansions for Large Variable

In this subsection we assume that none of a1,a2,,ap is a nonpositive integer.

Case p=q+1

The formal series (16.11.2) for Hq+1,q(z) converges if |z|>1, and

16.11.6 (=1q+1Γ(a)/=1qΓ(b))Fqq+1(a1,,aq+1b1,,bq;z)=Hq+1,q(-z),
|ph(-z)|π;

compare (16.8.8).

Case p=q

As z in |phz|π,

16.11.7 (=1qΓ(a)/=1qΓ(b))Fqq(a1,,aqb1,,bq;z)Hq,q(zπ)+Eq,q(z).

Here the upper or lower signs are chosen according as z lies in the upper or lower half-plane; in consequence, in the fractional powers (§4.2(iv)) of zeπi its phases are phzπ, respectively. (Either sign may be used when phz=0 since the first term on the right-hand side becomes exponentially small compared with the second term.)

For the special case a1=1, p=q=2 explicit representations for the right-hand side of (16.11.7) in terms of generalized hypergeometric functions are given in Kim (1972).

Case p=q-1

As z in |phz|π,

16.11.8 (=1q-1Γ(a)/=1qΓ(b))Fqq-1(a1,,aq-1b1,,bq;-z)Hq-1,q(z)+Eq-1,q(ze-π)+Eq-1,q(zeπ),

with the same conventions on the phases of zeπi.

Case pq-2

As z in |phz|π,

16.11.9 (=1pΓ(a)/=1qΓ(b))Fqp(a1,,apb1,,bq;-z)Ep,q(ze-π)+Ep,q(zeπ),

with the same conventions on the phases of zeπi.

§16.11(iii) Expansions for Large Parameters

If z is fixed and |ph(1-z)|<π, then for each nonnegative integer m

16.11.10 Fpp+1(a1+r,,ak-1+r,ak,,ap+1b1+r,,bk+r,bk+1,,bp;z)=n=0m-1(a1+r)n(ak-1+r)n(ak)n(ap+1)n(b1+r)n(bk+r)n(bk+1)n(bp)nznn!+O(1rm),

as r+. Here k can have any integer value from 1 to p. Also if p<q, then

16.11.11 Fqp(a1+r,,ap+rb1+r,,bq+r;z)=n=0m-1(a1+r)n(ap+r)n(b1+r)n(bq+r)nznn!+O(1r(q-p)m),

again as r+. For these and other results see Knottnerus (1960). See also Luke (1969a, §7.3).

Asymptotic expansions for the polynomials Fqp+2(-r,r+a0,a;b;z) as r through integer values are given in Fields and Luke (1963b, a) and Fields (1965).