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16 Generalized Hypergeometric Functions and Meijer G-FunctionGeneralized Hypergeometric Functions

§16.2 Definition and Analytic Properties

Contents

§16.2(i) Generalized Hypergeometric Series

Throughout this chapter it is assumed that none of the bottom parameters b1, b2, , bq is a nonpositive integer, unless stated otherwise. Then formally

16.2.1 Fqp(a1,,apb1,,bq;z)=k=0(a1)k(ap)k(b1)k(bq)kzkk!.

Equivalently, the function is denoted by Fqp(ab;z) or Fqp(a;b;z), and sometimes, for brevity, by Fqp(z).

§16.2(ii) Case pq

When pq the series (16.2.1) converges for all finite values of z and defines an entire function.

§16.2(iii) Case p=q+1

Suppose first one or more of the top parameters aj is a nonpositive integer. Then the series (16.2.1) terminates and the generalized hypergeometric function is a polynomial in z.

If none of the aj is a nonpositive integer, then the radius of convergence of the series (16.2.1) is 1, and outside the open disk |z|<1 the generalized hypergeometric function is defined by analytic continuation with respect to z. The branch obtained by introducing a cut from 1 to + on the real axis, that is, the branch in the sector |ph(1-z)|π, is the principal branch (or principal value) of Fqq+1(a;b;z); compare §4.2(i). Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at z=0,1, and . Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values.

On the circle |z|=1 the series (16.2.1) is absolutely convergent if γq>0, convergent except at z=1 if -1<γq0, and divergent if γq-1, where

16.2.2 γq=(b1++bq)-(a1++aq+1).

§16.2(iv) Case p>q+1

Polynomials

In general the series (16.2.1) diverges for all nonzero values of z. However, when one or more of the top parameters aj is a nonpositive integer the series terminates and the generalized hypergeometric function is a polynomial in z. Note that if -m is the value of the numerically largest aj that is a nonpositive integer, then the identity

16.2.3 Fqp+1(-m,ab;z)=(a)m(-z)m(b)mFpq+1(-m,1-m-b1-m-a;(-1)p+qz)

can be used to interchange p and q.

Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via

16.2.4 k=0m(a)k(b)kzkk!=(a)mzm(b)mm!Fpq+2(-m,1,1-m-b1-m-a;(-1)p+q+1z).

Non-Polynomials

See §16.5 for the definition of Fqp(a;b;z) as a contour integral when p>q+1 and none of the ak is a nonpositive integer. (However, except where indicated otherwise in the DLMF we assume that when p>q+1 at least one of the ak is a nonpositive integer.)

§16.2(v) Behavior with Respect to Parameters

Let

16.2.5 Fqp(a;b;z)=Fqp(a1,,apb1,,bq;z)/(Γ(b1)Γ(bq))=k=0(a1)k(ap)kΓ(b1+k)Γ(bq+k)zkk!;

compare (15.2.2) in the case p=2, q=1. When pq+1 and z is fixed and not a branch point, any branch of Fqp(a;b;z) is an entire function of each of the parameters a1,,ap,b1,,bq.