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15 Hypergeometric FunctionProperties

§15.2 Definitions and Analytical Properties


§15.2(i) Gauss Series

The hypergeometric function F(a,b;c;z) is defined by the Gauss series

15.2.1 F(a,b;c;z)=s=0(a)s(b)s(c)ss!zs=1+abcz+a(a+1)b(b+1)c(c+1)2!z2+=Γ(c)Γ(a)Γ(b)s=0Γ(a+s)Γ(b+s)Γ(c+s)s!zs,

on the disk |z|<1, and by analytic continuation elsewhere. In general, F(a,b;c;z) does not exist when c=0,-1,-2,. The branch obtained by introducing a cut from 1 to + on the real z-axis, that is, the branch in the sector |ph(1-z)|π, is the principal branch (or principal value) of F(a,b;c;z).

For all values of c

15.2.2 F(a,b;c;z)=s=0(a)s(b)sΓ(c+s)s!zs,

again with analytic continuation for other values of z, and with the principal branch defined in a similar way.

Except where indicated otherwise principal branches of F(a,b;c;z) and F(a,b;c;z) are assumed throughout the DLMF.

The difference between the principal branches on the two sides of the branch cut (§4.2(i)) is given by

15.2.3 F(a,bc;x+i0)-F(a,bc;x-i0)=2πiΓ(a)Γ(b)(x-1)c-a-bF(c-a,c-bc-a-b+1;1-x),

On the circle of convergence, |z|=1, the Gauss series:

  • (a)

    Converges absolutely when (c-a-b)>0.

  • (b)

    Converges conditionally when -1<(c-a-b)0 and z=1 is excluded.

  • (c)

    Diverges when (c-a-b)-1.

For the case z=1 see also §15.4(ii).

§15.2(ii) Analytic Properties

The principal branch of F(a,b;c;z) is an entire function of a, b, and c. The same is true of other branches, provided that z=0, 1, and are excluded. As a multivalued function of z, F(a,b;c;z) is analytic everywhere except for possible branch points at z=0, 1, and . The same properties hold for F(a,b;c;z), except that as a function of c, F(a,b;c;z) in general has poles at c=0,-1,-2,.

Because of the analytic properties with respect to a, b, and c, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. In particular

15.2.3_5 limc-nF(a,b;c;z)Γ(c)=F(a,b;-n;z)=(a)n+1(b)n+1(n+1)!zn+1F(a+n+1,b+n+1;n+2;z),

For example, when a=-m, m=0,1,2,, and c0,-1,-2,, F(a,b;c;z) is a polynomial:

15.2.4 F(-m,b;c;z)=n=0m(-m)n(b)n(c)nn!zn=n=0m(-1)n(mn)(b)n(c)nzn.

This formula is also valid when c=-m-, =0,1,2,, provided that we use the interpretation

15.2.5 F(-m,b-m-;z)=limc-m-(lima-mF(a,bc;z)),

and not

15.2.6 F(-m,b-m-;z)=lima-mF(a,ba-;z),

which sometimes needs to be used in §15.4. (Both interpretations give solutions of the hypergeometric differential equation (15.10.1), as does F(a,b;c;z), which is analytic at c=0,-1,-2,.)

For comparison of F(a,b;c;z) and F(a,b;c;z), with the former using the limit interpretation (15.2.5), see Figures 15.3.6 and 15.3.7.

Let m be a nonnegative integer. Formula (15.4.6) reads F(a,b;a;z)=(1-z)-b. The right-hand side can be seen as an analytical continuation for the left-hand side when a approaches -m. In that case we are using interpretation (15.2.6) since with interpretation (15.2.5) we would obtain that F(-m,b;-m;z) is equal to the first m+1 terms of the Maclaurin series for (1-z)-b.