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16 Generalized Hypergeometric Functions & Meijer G-FunctionGeneralized Hypergeometric Functions

Β§16.2 Definition and Analytic Properties

Contents
  1. Β§16.2(i) Generalized Hypergeometric Series
  2. Β§16.2(ii) Case p≀q
  3. Β§16.2(iii) Case p=q+1
  4. Β§16.2(iv) Case p>q+1
  5. Β§16.2(v) Behavior with Respect to Parameters

Β§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)=βˆ‘k=0∞(a1)k⁒⋯⁒(ap)k(b1)k⁒⋯⁒(bq)k⁒zkk!.

Equivalently, the function is denoted by Fqp⁑(πšπ›;z) or Fqp⁑(𝐚;𝐛;z), and sometimes, for brevity, by Fqp⁑(z).

Β§16.2(ii) Case p≀q

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

Β§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⁑(1βˆ’z)|≀π, is the principal branch (or principal value) of Fqq+1⁑(𝐚;𝐛;z); compare Β§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 β„œβ‘Ξ³q>0, convergent except at z=1 if βˆ’1<β„œβ‘Ξ³q≀0, and divergent if β„œβ‘Ξ³qβ‰€βˆ’1, where

16.2.2 Ξ³q=(b1+β‹―+bq)βˆ’(a1+β‹―+aq+1).

Β§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 βˆ’m is the value of the numerically largest aj that is a nonpositive integer, then the identity

16.2.3 Fqp+1⁑(βˆ’m,πšπ›;z)=(𝐚)m⁒(βˆ’z)m(𝐛)m⁒Fpq+1⁑(βˆ’m,1βˆ’mβˆ’π›1βˆ’mβˆ’πš;(βˆ’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 βˆ‘k=0m(𝐚)k(𝐛)k⁒zkk!=(𝐚)m⁒zm(𝐛)m⁒m!⁒Fpq+2⁑(βˆ’m,1,1βˆ’mβˆ’π›1βˆ’mβˆ’πš;(βˆ’1)p+q+1z).

Non-Polynomials

See Β§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.)

Β§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))=βˆ‘k=0∞(a1)k⁒⋯⁒(ap)kΓ⁑(b1+k)⁒⋯⁒Γ⁑(bq+k)⁒zkk!;

compare (15.2.2) in the case p=2, q=1. When p≀q+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.