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16 Generalized Hypergeometric Functions & Meijer G-FunctionGeneralized Hypergeometric Functions

Β§16.11 Asymptotic Expansions

  1. Β§16.11(i) Formal Series
  2. Β§16.11(ii) Expansions for Large Variable
  3. Β§16.11(iii) Expansions for Large Parameters

Β§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)=(2⁒π)(pβˆ’q)/2β’ΞΊβˆ’Ξ½βˆ’(1/2)⁒eκ⁒z1/ΞΊβ’βˆ‘k=0∞ck⁒(κ⁒z1/ΞΊ)Ξ½βˆ’k,
16.11.2 Hp,q⁑(z)=βˆ‘m=1pβˆ‘k=0∞(βˆ’1)kk!⁒Γ⁑(am+k)⁒(βˆβ„“=1β„“β‰ mpΓ⁑(aβ„“βˆ’amβˆ’k)/βˆβ„“=1qΓ⁑(bβ„“βˆ’amβˆ’k))⁒zβˆ’amβˆ’k.

In (16.11.1)

16.11.3 ΞΊ =qβˆ’p+1,
Ξ½ =a1+β‹―+apβˆ’b1βˆ’β‹―βˆ’bq+12⁒(qβˆ’p),


16.11.4 c0 =1,
ck =βˆ’1kβ’ΞΊΞΊβ’βˆ‘m=0kβˆ’1cm⁒ek,m,


16.11.5 ek,m=βˆ‘j=1q+1(1βˆ’Ξ½βˆ’ΞΊβ’bj+m)ΞΊ+kβˆ’m⁒(βˆβ„“=1p(aβ„“βˆ’bj)/βˆβ„“=1β„“β‰ jq+1(bβ„“βˆ’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=βˆ’aj,βˆ’ajβˆ’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.)

Β§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

Case p=q

As zβ†’βˆž in |ph⁑z|≀π,

16.11.7 (βˆβ„“=1qΓ⁑(aβ„“)/βˆβ„“=1qΓ⁑(bβ„“))⁒Fqq⁑(a1,…,aqb1,…,bq;z)∼Hq,q⁑(z⁒eβˆ“Ο€β’i)+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 (Β§4.2(iv)) of z⁒eβˆ“Ο€β’i its phases are ph⁑zβˆ“Ο€, respectively. (Either sign may be used when ph⁑z=0 since the first term on the right-hand side becomes exponentially small compared with the second term.)

Explicit representations for the coefficients ck are given in Volkmer and Wood (2014). The special case a1=1, p=q=2 is discussed in Kim (1972).

Case p=qβˆ’1

As zβ†’βˆž in |ph⁑z|≀π,

16.11.8 (βˆβ„“=1qβˆ’1Γ⁑(aβ„“)/βˆβ„“=1qΓ⁑(bβ„“))⁒Fqqβˆ’1⁑(a1,…,aqβˆ’1b1,…,bq;βˆ’z)∼Hqβˆ’1,q⁑(z)+Eqβˆ’1,q⁑(z⁒eβˆ’Ο€β’i)+Eqβˆ’1,q⁑(z⁒eπ⁒i),

with the same conventions on the phases of z⁒eβˆ“Ο€β’i.

Case p≀qβˆ’2

As zβ†’βˆž in |ph⁑z|≀π,

16.11.9 (βˆβ„“=1pΓ⁑(aβ„“)/βˆβ„“=1qΓ⁑(bβ„“))⁒Fqp⁑(a1,…,apb1,…,bq;βˆ’z)∼Ep,q⁑(z⁒eβˆ’Ο€β’i)+Ep,q⁑(z⁒eπ⁒i),

with the same conventions on the phases of z⁒eβˆ“Ο€β’i.

Β§16.11(iii) Expansions for Large Parameters

If z is fixed and |ph⁑(1βˆ’z)|<Ο€, then for each nonnegative integer m

16.11.10 Fpp+1⁑(a1+r,…,akβˆ’1+r,ak,…,ap+1b1+r,…,bk+r,bk+1,…,bp;z)=βˆ‘n=0mβˆ’1(a1+r)n⁒⋯⁒(akβˆ’1+r)n⁒(ak)n⁒⋯⁒(ap+1)n(b1+r)n⁒⋯⁒(bk+r)n⁒(bk+1)n⁒⋯⁒(bp)n⁒znn!+O⁑(1rm),

as rβ†’+∞. 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)=βˆ‘n=0mβˆ’1(a1+r)n⁒⋯⁒(ap+r)n(b1+r)n⁒⋯⁒(bq+r)n⁒znn!+O⁑(1r(qβˆ’p)⁒m),

again as rβ†’+∞. For these and other results see Knottnerus (1960). See also Luke (1969a, Β§7.3).

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