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16 Generalized Hypergeometric Functions and Meijer G-FunctionGeneralized Hypergeometric Functions

§16.3 Derivatives and Contiguous Functions

Contents

§16.3(i) Differentiation Formulas

16.3.1 nznFqp(a1,,apb1,,bq;z)=(a)n(b)nFqp(a1+n,,ap+nb1+n,,bq+n;z),
16.3.2 nzn(zγFqp(a1,,apb1,,bq;z))=(γ-n+1)nzγ-nFq+1p+1(γ+1,a1,,apγ+1-n,b1,,bq;z),
16.3.3 (zzz)n(zγ-1Fqp+1(γ,a1,,apb1,,bq;z))=(γ)nzγ+n-1Fqp+1(γ+n,a1,,apb1,,bq;z),
16.3.4 nzn(zγ-1Fq+1p(a1,,apγ,b1,,bq;z))=(γ-n)nzγ-n-1Fq+1p(a1,,apγ-n,b1,,bq;z).

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

16.3.5 (zzz)n=znnznzn,
n=1,2,.

§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 pq+1, then any q+2 distinct contiguous functions are linearly related. Examples are provided by the following recurrence relations:

16.3.6 zF10(-;b+1;z)+b(b-1)F10(-;b;z)-b(b-1)F10(-;b-1;z)=0,
16.3.7 F23(a1+2,a2,a3b1,b2;z)a1(a1+1)(1-z)+F23(a1+1,a2,a3b1,b2;z)a1(b1+b2-3a1-2+z(2a1-a2-a3+1))+F23(a1,a2,a3b1,b2;z)((2a1-b1)(2a1-b2)+a1-a12-z(a1-a2)(a1-a3))-F23(a1-1,a2,a3b1,b2;z)(a1-b1)(a1-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).