Digital Library of Mathematical Functions
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15 Hypergeometric FunctionProperties

§15.5 Derivatives and Contiguous Functions

Contents

§15.5(i) Differentiation Formulas

15.5.1 zF(a,b;c;z)=abcF(a+1,b+1;c+1;z),
15.5.2 nznF(a,b;c;z)=(a)n(b)n(c)nF(a+n,b+n;c+n;z).
15.5.3 (zzz)n(za-1F(a,b;c;z))=(a)nza+n-1F(a+n,b;c;z).
15.5.4 nzn(zc-1F(a,b;c;z))=(c-n)nzc-n-1F(a,b;c-n;z).
15.5.5 (zzz)n(zc-a-1(1-z)a+b-cF(a,b;c;z))=(c-a)nzc-a+n-1(1-z)a-n+b-cF(a-n,b;c;z).
15.5.6 nzn((1-z)a+b-cF(a,b;c;z))=(c-a)n(c-b)n(c)n(1-z)a+b-c-nF(a,b;c+n;z).
15.5.7 ((1-z)z(1-z))n((1-z)a-1F(a,b;c;z))=(-1)n(a)n(c-b)n(c)n(1-z)a+n-1F(a+n,b;c+n;z).
15.5.8 ((1-z)z(1-z))n(zc-1(1-z)b-cF(a,b;c;z))=(c-n)nzc-n-1(1-z)b-c+nF(a-n,b;c-n;z).
15.5.9 nzn(zc-1(1-z)a+b-cF(a,b;c;z))=(c-n)nzc-n-1(1-z)a+b-c-nF(a-n,b-n;c-n;z).

Other versions of several of the identities in this subsection can be constructed with the aid of the operator identity

15.5.10 (zzz)n=znnznzn,
n=1,2,3,.

See Erdélyi et al. (1953a, pp. 102–103).

§15.5(ii) Contiguous Functions

The six functions F(a±1,b;c;z), F(a,b±1;c;z), F(a,b;c±1;z) are said to be contiguous to F(a,b;c;z).

15.5.11 (c-a)F(a-1,b;c;z)+(2a-c+(b-a)z)F(a,b;c;z)+a(z-1)F(a+1,b;c;z) =0,
15.5.12 (b-a)F(a,b;c;z)+aF(a+1,b;c;z)-bF(a,b+1;c;z) =0,
15.5.13 (c-a-b)F(a,b;c;z)+a(1-z)F(a+1,b;c;z)-(c-b)F(a,b-1;c;z) =0,
15.5.14 c(a+(b-c)z)F(a,b;c;z)-ac(1-z)F(a+1,b;c;z)+(c-a)(c-b)zF(a,b;c+1;z) =0,
15.5.15 (c-a-1)F(a,b;c;z)+aF(a+1,b;c;z)-(c-1)F(a,b;c-1;z) =0,
15.5.16 c(1-z)F(a,b;c;z)-cF(a-1,b;c;z)+(c-b)zF(a,b;c+1;z) =0,
15.5.17 (a-1+(b+1-c)z)F(a,b;c;z)+(c-a)F(a-1,b;c;z)-(c-1)(1-z)F(a,b;c-1;z) =0,
15.5.18 c(c-1)(z-1)F(a,b;c-1;z)+c(c-1-(2c-a-b-1)z)F(a,b;c;z)+(c-a)(c-b)zF(a,b;c+1;z) =0.

By repeated applications of (15.5.11)–(15.5.18) any function F(a+k,b+;c+m;z), in which k,,m are integers, can be expressed as a linear combination of F(a,b;c;z) and any one of its contiguous functions, with coefficients that are rational functions of a,b,c, and z.

An equivalent equation to the hypergeometric differential equation (15.10.1) is

15.5.19 z(1-z)(a+1)(b+1)F(a+2,b+2;c+2;z)+(c-(a+b+1)z)(c+1)F(a+1,b+1;c+1;z)-c(c+1)F(a,b;c;z)=0.

Further contiguous relations include:

15.5.20 z(1-z)(F(a,b;c;z)/z)=(c-a)F(a-1,b;c;z)+(a-c+bz)F(a,b;c;z)=(c-b)F(a,b-1;c;z)+(b-c+az)F(a,b;c;z),
15.5.21 c(1-z)(F(a,b;c;z)/z)=(c-a)(c-b)F(a,b;c+1;z)+c(a+b-c)F(a,b;c;z).