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15 Hypergeometric FunctionProperties

§15.9 Relations to Other Functions

Contents

§15.9(i) Orthogonal Polynomials

For the notation see §§18.3 and 18.19.

Jacobi

15.9.1 Pn(α,β)(x)=(α+1)nn!F(-n,n+α+β+1α+1;1-x2).

Gegenbauer (or Ultraspherical)

Chebyshev

Legendre

Krawtchouk

15.9.8 Kn(x;p,N)=F(-n,-x-N;1p),
n=0,1,2,,N;

compare also §15.2(ii).

Meixner

Meixner–Pollaczek

§15.9(ii) Jacobi Function

This is a generalization of Jacobi polynomials (§18.3) and has the representation

15.9.11 ϕλ(α,β)(t)=F(12(α+β+1-iλ),12(α+β+1+iλ)α+1;-sinh2t).

The Jacobi transform is defined as

15.9.12 f~(λ)=0f(t)ϕλ(α,β)(t)(2sinht)2α+1(2cosht)2β+1dt,

with inverse

15.9.13 f(t)=12πi-iif~(iλ)Φiλ(α,β)(t)Γ(12(α+β+1+λ))Γ(12(α-β+1+λ))Γ(α+1)Γ(λ)2α+β+1-λdλ,

where the contour of integration is located to the right of the poles of the gamma functions in the integrand, and

15.9.14 Φλ(α,β)(t)=(2cosht)iλ-α-β-1F(12(α+β+1-iλ),12(α-β+1-iλ)1-iλ;sech2t).

For this result, together with restrictions on the functions f(t) and f~(λ), see Koornwinder (1984a).

§15.9(iii) Gegenbauer Function

This is a generalization of Gegenbauer (or ultraspherical) polynomials (§18.3). It is defined by:

15.9.15 Cα(λ)(z)=Γ(α+2λ)Γ(2λ)Γ(α+1)F(-α,α+2λλ+12;1-z2).

§15.9(iv) Associated Legendre Functions; Ferrers Functions

Any hypergeometric function for which a quadratic transformation exists can be expressed in terms of associated Legendre functions or Ferrers functions. For examples see §§14.3(i)14.3(iii) and 14.21(iii).

The following formulas apply with principal branches of the hypergeometric functions, associated Legendre functions, and fractional powers.

15.9.16 F(a,b2b;z) =πΓ(b)z-b+(1/2)(1-z)(b-a-(1/2))/2Pa-b-(1/2)-b+(1/2)(2-z21-z),
b0,-1,-2,, |ph(1-z)|<π and |1-z|<1.
15.9.17 F(a,a+12c;z) =2c-1z(1-c)/2(1-z)-a+((c-1)/2)P2a-c1-c(11-z),
|phz|<π and |ph(1-z)|<π.
15.9.18 F(a,ba+b+12;z) =2a+b-(1/2)(-z)(-a-b+(1/2))/2Pa-b-(1/2)-a-b+(1/2)(1-z),
|ph(-z)|<π.
15.9.19 F(a,ba-b+1;z) =z(b-a)/2(1-z)-bP-bb-a(1+z1-z),
|phz|<π and |ph(1-z)|<π.
15.9.20 F(a,b12(a+b+1);z) =(-z(1-z))(1-a-b)/4P(a-b-1)/2(1-a-b)/2(1-2z),
|ph(-z)|<π.
15.9.21 F(a,1-ac;z) =(-z1-z)(1-c)/2P-a1-c(1-2z),
|ph(-z)|<π.
For the case 0<z<1 see (14.3.1).
15.9.22 F(a,b12;z)=2a+b-(3/2)πΓ(a+12)Γ(b+12)(z-1)(-a-b+(1/2))/2×(e±πi(a+b-(1/2))Pa-b-(1/2)-a-b+(1/2)(-z)+Pa-b-(1/2)-a-b+(1/2)(z)),
a,b-12,-32,-52,, 0<|phz|<π,

where the sign in the exponential is ± according as z0.

15.9.23 F(a,b32;z)=2a+b-(5/2)πzΓ(a-12)Γ(b-12)(z-1)(-a-b+(3/2))/2×(e±πi(a+b-(3/2))Pa-b-(1/2)-a-b+(3/2)(-z)-Pa-b-(1/2)-a-b+(3/2)(z)),
a,b12,-12,-32,, 0<|phz|<π,

where the sign in the exponential is ± according as z0.