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15 Hypergeometric FunctionProperties

§15.5 Derivatives and Contiguous Functions

Contents
  1. §15.5(i) Differentiation Formulas
  2. §15.5(ii) Contiguous Functions

§15.5(i) Differentiation Formulas

15.5.1 ddzF(a,b;c;z)=abcF(a+1,b+1;c+1;z),
15.5.2 dndznF(a,b;c;z)=(a)n(b)n(c)nF(a+n,b+n;c+n;z).
15.5.3 (zddzz)n(za1F(a,b;c;z))=(a)nza+n1F(a+n,b;c;z).
15.5.4 dndzn(zc1F(a,b;c;z))=(cn)nzcn1F(a,b;cn;z).
15.5.5 (zddzz)n(zca1(1z)a+bcF(a,b;c;z))=(ca)nzca+n1(1z)an+bcF(an,b;c;z).
15.5.6 dndzn((1z)a+bcF(a,b;c;z))=(ca)n(cb)n(c)n(1z)a+bcnF(a,b;c+n;z).
15.5.7 ((1z)ddz(1z))n((1z)a1F(a,b;c;z))=(1)n(a)n(cb)n(c)n(1z)a+n1F(a+n,b;c+n;z).
15.5.8 ((1z)ddz(1z))n(zc1(1z)bcF(a,b;c;z))=(cn)nzcn1(1z)bc+nF(an,b;cn;z).
15.5.9 dndzn(zc1(1z)a+bcF(a,b;c;z))=(cn)nzcn1(1z)a+bcnF(an,bn;cn;z).

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

15.5.10 (zddzz)n=zndndznzn,
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 (ca)F(a1,b;c;z)+(2ac+(ba)z)F(a,b;c;z)+a(z1)F(a+1,b;c;z) =0,
15.5.12 (ba)F(a,b;c;z)+aF(a+1,b;c;z)bF(a,b+1;c;z) =0,
15.5.13 (cab)F(a,b;c;z)+a(1z)F(a+1,b;c;z)(cb)F(a,b1;c;z) =0,
15.5.14 c(a+(bc)z)F(a,b;c;z)ac(1z)F(a+1,b;c;z)+(ca)(cb)zF(a,b;c+1;z) =0,
15.5.15 (ca1)F(a,b;c;z)+aF(a+1,b;c;z)(c1)F(a,b;c1;z) =0,
15.5.16 c(1z)F(a,b;c;z)cF(a1,b;c;z)+(cb)zF(a,b;c+1;z) =0,
15.5.16_5 F(a,b;c;z)F(a1,b;c;z)(b/c)zF(a,b+1;c+1;z) =0,
15.5.17 (a1+(b+1c)z)F(a,b;c;z)+(ca)F(a1,b;c;z)(c1)(1z)F(a,b;c1;z) =0,
15.5.18 c(c1)(z1)F(a,b;c1;z)+c(c1(2cab1)z)F(a,b;c;z)+(ca)(cb)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(1z)(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(1z)(dF(a,b;c;z)/dz)=(ca)F(a1,b;c;z)+(ac+bz)F(a,b;c;z)=(cb)F(a,b1;c;z)+(bc+az)F(a,b;c;z),
15.5.21 c(1z)(dF(a,b;c;z)/dz)=(ca)(cb)F(a,b;c+1;z)+c(a+bc)F(a,b;c;z).