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16 Generalized Hypergeometric Functions & Meijer G-FunctionGeneralized Hypergeometric Functions

Β§16.8 Differential Equations

  1. Β§16.8(i) Classification of Singularities
  2. Β§16.8(ii) The Generalized Hypergeometric Differential Equation
  3. Β§16.8(iii) Confluence of Singularities

Β§16.8(i) Classification of Singularities

An ordinary point of the differential equation

16.8.1 dnwdzn+fnβˆ’1⁑(z)⁒dnβˆ’1wdznβˆ’1+fnβˆ’2⁑(z)⁒dnβˆ’2wdznβˆ’2+β‹―+f1⁑(z)⁒dwdz+f0⁑(z)⁒w=0

is a value z0 of z at which all the coefficients fj⁑(z), j=0,1,…,nβˆ’1, are analytic. If z0 is not an ordinary point but (zβˆ’z0)nβˆ’j⁒fj⁑(z), j=0,1,…,nβˆ’1, are analytic at z=z0, then z0 is a regular singularity. All other singularities are irregular. Compare Β§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 Β§2.7(i).

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

Β§16.8(ii) The Generalized Hypergeometric Differential Equation

With the notation

16.8.2 𝐷 =ddz,
Ο‘ =z⁒ddz,

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

16.8.3 (Ο‘(Ο‘+b1βˆ’1)⁒⋯⁒(Ο‘+bqβˆ’1)βˆ’z⁒(Ο‘+a1)⁒⋯⁒(Ο‘+ap))⁒w=0.


16.8.4 zq⁒𝐷q+1w+βˆ‘j=1qzjβˆ’1⁒(Ξ±j⁒z+Ξ²j)⁒𝐷jw+Ξ±0⁒w=0,


16.8.5 zq⁒(1βˆ’z)⁒𝐷q+1w+βˆ‘j=1qzjβˆ’1⁒(Ξ±j⁒z+Ξ²j)⁒𝐷jw+Ξ±0⁒w=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) =z1βˆ’bj⁒Fqp⁑(1+a1βˆ’bj,…,1+apβˆ’bj2βˆ’bj,1+b1βˆ’bj,β€¦βˆ—β€¦,1+bqβˆ’bj;z),

where βˆ— indicates that the entry 1+bjβˆ’bj is omitted. For other values of the bj, series solutions in powers of z (possibly involving also ln⁑z) can be constructed via a limiting process; compare Β§2.7(i) in the case of second-order differential equations. For details see Smith (1939a, b), and NΓΈrlund (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)=(βˆ’z)βˆ’aj⁒Fqq+1⁑(aj,1βˆ’b1+aj,…,1βˆ’bq+aj1βˆ’a1+aj,β€¦βˆ—β€¦,1βˆ’aq+1+aj;1z),

where βˆ— indicates that the entry 1βˆ’aj+aj is omitted. We have the connection formula

16.8.8 Fqq+1⁑(a1,…,aq+1b1,…,bq;z)=βˆ‘j=1q+1(∏k=1kβ‰ jq+1Γ⁑(akβˆ’aj)Γ⁑(ak)/∏k=1qΓ⁑(bkβˆ’aj)Γ⁑(bk))⁒w~j⁒(z),

More generally if z0 (βˆˆβ„‚) is an arbitrary constant, |zβˆ’z0|>max⁑(|z0|,|z0βˆ’1|), and |ph⁑(z0βˆ’z)|<Ο€, then

16.8.9 (∏k=1q+1Γ⁑(ak)/∏k=1qΓ⁑(bk))⁒Fqq+1⁑(a1,…,aq+1b1,…,bq;z)=βˆ‘j=1q+1(z0βˆ’z)βˆ’ajβ’βˆ‘n=0βˆžΞ“β‘(aj+n)n!⁒(∏k=1kβ‰ jq+1Γ⁑(akβˆ’ajβˆ’n)/∏k=1qΓ⁑(bkβˆ’ajβˆ’n))Γ—Fqq+1⁑(a1βˆ’ajβˆ’n,…,aq+1βˆ’ajβˆ’nb1βˆ’ajβˆ’n,…,bqβˆ’ajβˆ’n;z0)⁒(zβˆ’z0)βˆ’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 NΓΈrlund (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 BΓΌhring (1987b) for the case q=2, and in BΓΌhring (1992) for the general case.

Β§16.8(iii) Confluence of Singularities

If p≀q, then

16.8.10 lim|Ξ±|β†’βˆžFqp+1⁑(a1,…,ap,Ξ±b1,…,bq;zΞ±)=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 p≀q+1 and |ph⁑β|β‰€Ο€βˆ’Ξ΄ (<Ο€), then

16.8.11 lim|Ξ²|β†’βˆžFq+1p⁑(a1,…,apb1,…,bq,Ξ²;β⁒z)=Fqp⁑(a1,…,apb1,…,bq;z),

provided that in the case p=q+1 we have |z|<1 when |ph⁑β|≀12⁒π, and |z|<|sin⁑(ph⁑β)| when 12⁒π≀|ph⁑β|β‰€Ο€βˆ’Ξ΄ (<Ο€).