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16 Generalized Hypergeometric Functions & Meijer G-FunctionGeneralized Hypergeometric Functions

Β§16.3 Derivatives and Contiguous Functions

Contents
  1. Β§16.3(i) Differentiation Formulas
  2. Β§16.3(ii) Contiguous Functions

Β§16.3(i) Differentiation Formulas

16.3.1 dndzn⁑Fqp⁑(a1,…,apb1,…,bq;z)=(a)n(b)n⁒Fqp⁑(a1+n,…,ap+nb1+n,…,bq+n;z),
16.3.2 dndzn⁑(zγ⁒Fqp⁑(a1,…,apb1,…,bq;z))=(Ξ³βˆ’n+1)n⁒zΞ³βˆ’n⁒Fq+1p+1⁑(Ξ³+1,a1,…,apΞ³+1βˆ’n,b1,…,bq;z),
16.3.3 (z⁒ddz⁑z)n⁒(zΞ³βˆ’1⁒Fqp+1⁑(Ξ³,a1,…,apb1,…,bq;z))=(Ξ³)n⁒zΞ³+nβˆ’1⁒Fqp+1⁑(Ξ³+n,a1,…,apb1,…,bq;z),
16.3.4 dndzn⁑(zΞ³βˆ’1⁒Fq+1p⁑(a1,…,apΞ³,b1,…,bq;z))=(Ξ³βˆ’n)n⁒zΞ³βˆ’nβˆ’1⁒Fq+1p⁑(a1,…,apΞ³βˆ’n,b1,…,bq;z).

Other versions of these identities can be constructed with the aid of the operator identity

16.3.5 (z⁒ddz⁑z)n=zn⁒dndzn⁑zn,
n=1,2,….

Β§16.3(ii) Contiguous Functions

Two generalized hypergeometric functions Fqp⁑(a;b;z) are (generalized) contiguous if they have the same pair of values of p and q, and corresponding parameters differ by integers. If p≀q+1, then any q+2 distinct contiguous functions are linearly related. Examples are provided by the following recurrence relations:

16.3.6 z⁒F10⁑(βˆ’;b+1;z)+b⁒(bβˆ’1)⁒F10⁑(βˆ’;b;z)βˆ’b⁒(bβˆ’1)⁒F10⁑(βˆ’;bβˆ’1;z)=0,
16.3.7 F23⁑(a1+2,a2,a3b1,b2;z)⁒a1⁒(a1+1)⁒(1βˆ’z)+F23⁑(a1+1,a2,a3b1,b2;z)⁒a1⁒(b1+b2βˆ’3⁒a1βˆ’2+z⁒(2⁒a1βˆ’a2βˆ’a3+1))+F23⁑(a1,a2,a3b1,b2;z)⁒((2⁒a1βˆ’b1)⁒(2⁒a1βˆ’b2)+a1βˆ’a12βˆ’z⁒(a1βˆ’a2)⁒(a1βˆ’a3))βˆ’F23⁑(a1βˆ’1,a2,a3b1,b2;z)⁒(a1βˆ’b1)⁒(a1βˆ’b2)=0.

For further examples see §§13.3(i), 15.5(ii), and the following references: Rainville (1960, §48), Wimp (1968), and Luke (1975, §5.13).