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19 Elliptic IntegralsSymmetric Integrals

§19.19 Taylor and Related Series

For N=0,1,2, define the homogeneous hypergeometric polynomial

19.19.1 TN(b,z)=(b1)m1(bn)mnm1!mn!z1m1znmn,

where the summation extends over all nonnegative integers m1,,mn whose sum is N. The following two multivariate hypergeometric series apply to each of the integrals (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23):

19.19.2 R-a(b;z)=N=0(a)N(c)NTN(b,1-z),
c=j=1nbj, |1-zj|<1,
19.19.3 R-a(b;z)=zn-aN=0(a)N(c)NTN(b1,,bn-1;1-(z1/zn),,1-(zn-1/zn)),
c=j=1nbj, |1-(zj/zn)|<1.

If n=2, then (19.19.3) is a Gauss hypergeometric series (see (19.25.43) and (15.2.1)).

Define the elementary symmetric function Es(z) by

19.19.4 j=1n(1+tzj)=s=0ntsEs(z),

and define the n-tuple 12=(12,,12). Then

19.19.5 TN(12,z)=(-1)M+N(12)ME1m1(z)Enmn(z)m1!mn!,

where M=j=1nmj and the summation extends over all nonnegative integers m1,,mn such that j=1njmj=N.

This form of TN can be applied to (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) if we use

19.19.6 RJ(x,y,z,p)=R-32(12,12,12,12,12;x,y,z,p,p)

as well as (19.16.5) and (19.16.6). The number of terms in TN can be greatly reduced by using variables Z=1-(z/A) with A chosen to make E1(Z)=0. Then TN has at most one term if N5 in the series for RF. For RJ and RD, TN has at most one term if N3, and two terms if N=4 or 5.

19.19.7 R-a(12;z)=A-aN=0(a)N(12n)NTN(12,Z),


19.19.8 A =1nj=1nzj,
Zj =1-(zj/A),
E1(Z) =0,

Special cases are given in (19.36.1) and (19.36.2).