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16 Generalized Hypergeometric Functions & Meijer G-FunctionGeneralized Hypergeometric Functions

§16.8 Differential Equations


§16.8(i) Classification of Singularities

An ordinary point of the differential equation

16.8.1 dnwdzn+fn-1(z)dn-1wdzn-1+fn-2(z)dn-2wdzn-2++f1(z)dwdz+f0(z)w=0

is a value z0 of z at which all the coefficients fj(z), j=0,1,,n-1, are analytic. If z0 is not an ordinary point but (z-z0)n-jfj(z), j=0,1,,n-1, are analytic at z=z0, then z0 is a regular singularity. All other singularities are irregular. Compare §2.7(i) in the case n=2. Similar definitions apply in the case z0=: we transform into the origin by replacing z in (16.8.1) by 1/z; again compare §2.7(i).

For further information see Hille (1976, pp. 360–370).

§16.8(ii) The Generalized Hypergeometric Differential Equation

With the notation

16.8.2 D =ddz,
ϑ =zddz,

the function w=Fqp(a;b;z) satisfies the differential equation

16.8.3 (ϑ(ϑ+b1-1)(ϑ+bq-1)-z(ϑ+a1)(ϑ+ap))w=0.


16.8.4 zqDq+1w+j=1qzj-1(αjz+βj)Djw+α0w=0,


16.8.5 zq(1-z)Dq+1w+j=1qzj-1(αjz+βj)Djw+α0w=0,

where αj and βj are constants. Equation (16.8.4) has a regular singularity at z=0, and an irregular singularity at z=, whereas (16.8.5) has regular singularities at z=0, 1, and . In each case there are no other singularities. Equation (16.8.3) is of order max(p,q+1). In Letessier et al. (1994) examples are discussed in which the generalized hypergeometric function satisfies a differential equation that is of order 1 or even 2 less than might be expected.

When no bj is an integer, and no two bj differ by an integer, a fundamental set of solutions of (16.8.3) is given by

16.8.6 w0(z) =Fqp(a1,,apb1,,bq;z),
wj(z) =z1-bjFqp(1+a1-bj,,1+ap-bj2-bj,1+b1-bj,*,1+bq-bj;z),

where * indicates that the entry 1+bj-bj is omitted. For other values of the bj, series solutions in powers of z (possibly involving also lnz) can be constructed via a limiting process; compare §2.7(i) in the case of second-order differential equations. For details see Smith (1939a, b), and Nørlund (1955).

When p=q+1, and no two aj differ by an integer, another fundamental set of solutions of (16.8.3) is given by

16.8.7 w~j(z)=(-z)-ajFqq+1(aj,1-b1+aj,,1-bq+aj1-a1+aj,*,1-aq+1+aj;1z),

where * indicates that the entry 1-aj+aj is omitted. We have the connection formula

16.8.8 Fqq+1(a1,,aq+1b1,,bq;z)=j=1q+1(k=1kjq+1Γ(ak-aj)Γ(ak)/k=1qΓ(bk-aj)Γ(bk))w~j(z),

More generally if z0 () is an arbitrary constant, |z-z0|>max(|z0|,|z0-1|), and |ph(z0-z)|<π, then

16.8.9 (k=1q+1Γ(ak)/k=1qΓ(bk))Fqq+1(a1,,aq+1b1,,bq;z)=j=1q+1(z0-z)-ajn=0Γ(aj+n)n!(k=1kjq+1Γ(ak-aj-n)/k=1qΓ(bk-aj-n))×Fqq+1(a1-aj-n,,aq+1-aj-nb1-aj-n,,bq-aj-n;z0)(z-z0)-n.

(Note that the generalized hypergeometric functions on the right-hand side are polynomials in z0.)

When p=q+1 and some of the aj differ by an integer a limiting process can again be applied. For details see Nørlund (1955). In this reference it is also explained that in general when q>1 no simple representations in terms of generalized hypergeometric functions are available for the fundamental solutions near z=1. Analytical continuation formulas for Fqq+1(a;b;z) near z=1 are given in Bühring (1987b) for the case q=2, and in Bühring (1992) for the general case.

§16.8(iii) Confluence of Singularities

If pq, then

16.8.10 lim|α|Fqp+1(a1,,ap,αb1,,bq;zα)=Fqp(a1,,apb1,,bq;z).

Thus in the case p=q the regular singularities of the function on the left-hand side at α and coalesce into an irregular singularity at .

Next, if pq+1 and |phβ|π-δ (<π), then

16.8.11 lim|β|Fq+1p(a1,,apb1,,bq,β;βz)=Fqp(a1,,apb1,,bq;z),

provided that in the case p=q+1 we have |z|<1 when |phβ|12π, and |z|<|sin(phβ)| when 12π|phβ|π-δ (<π).