Hypergeometric Differential Equation - Gauss' Contiguous Relations

Gauss' Contiguous Relations

The six functions

are called contiguous to 2F1(a,b;c;z). Gauss showed that 2F1(a,b;c;z) can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of a,b,c, and z. This gives (6
2)=15 relations, given by identifying any two lines on the right hand side of

\begin{align} z\frac{dF}{dz} = z\frac{ab}{c}F(a+,b+,c+) &=a(F(a+)-F)\\ &=b(F(b+)-F)\\ &=(c-1)(F(c-)-F)\\ &=\frac{(c-a)F(a-)+(a-c+bz)F}{1-z}\\ &=\frac{(c-b)F(b-)+(b-c+az)F}{1-z}\\ &=z\frac{(c-a)(c-b)F(c+)+c(a+b-c)F}{c(1-z)}\\ \end{align}

In the notation above, and so on.

Repeatedly applying these relations gives a linear relation over C(z) between any three functions of the form

where m, n, and l are integers.

Read more about this topic:  Hypergeometric Differential Equation

Famous quotes containing the words contiguous and/or relations:

    “Now in contiguous drops the flood comes down,
    Threat’ning with deluge this devoted town.
    To shops in crowds the daggled females fly,
    Pretend to cheapen goods, but nothing buy.”
    —Jonathan Swift (1667–1745)

    “I want relations which are not purely personal, based on purely personal qualities; but relations based upon some unanimous accord in truth or belief, and a harmony of purpose, rather than of personality. I am weary of personality.... Let us be easy and impersonal, not forever fingering over our own souls, and the souls of our acquaintances, but trying to create a new life, a new common life, a new complete tree of life from the roots that are within us.”
    —D.H. (David Herbert)