Special functions Hypergeometric functions Theorem

Contiguous relations

Builds on Hypergeometric function Hypergeometric series

Statement

Let $a, b, c \in \C$ with $c \notin \set{1, 0, -1, -2, \ldots}$ . Write $F = {}_2F_1(a, b; c; z)$ for the hypergeometric function. Write $F(a \pm 1)$ , $F(b \pm 1)$ , $F(c \pm 1)$ for the six contiguous functions obtained by changing one parameter by $\pm 1$ .

(Gauss) Let $G$ and $H$ be two of the six contiguous functions. Then $F$ , $G$ , $H$ satisfy a linear relation whose coefficients are polynomials in $a, b, c, z$ , not all zero.
In particular, for all $z \in \C \setminus [1, \infty)$ , $(c - a) \, F(a - 1) + \big(2a - c + (b - a) z\big) F + a (z - 1) \, F(a + 1) = 0,$ $(a - b) \, F = a \, F(a + 1) - b \, F(b + 1).$

Remarks

The second relation in part 2 can be checked on the hypergeometric series directly. Write $t_n = \frac{(a)_n (b)_n}{(c)_n \, n!}$ for the coefficients of $F$ . Since $a \, (a+1)_n = (a)_n (a + n)$ , the coefficient of $z^n$ in $a F(a + 1)$ is $t_n (a + n)$ , and in the same way the coefficient in $b F(b + 1)$ is $t_n (b + n)$ . Their difference is $t_n (a - b)$ , the coefficient of $(a - b) F$ . The other relations follow by similar coefficient comparisons, and they extend from $\abs{z} < 1$ to the cut plane by analytic continuation. There are $\binom{6}{2} = 15$ such relations, and Gauss listed them all.

Example: for $a = 1$ , $b = 2$ the second relation gives $-F(1, 2; c; z) = F(2, 2; c; z) - 2 F(1, 3; c; z)$ , a relation between three explicit series.

By combining contiguous relations, for generic parameters the function $F(a + m, b + n; c + l; z)$ for integers $m, n, l$ is a combination of $F$ and one contiguous function, with coefficients rational in $a, b, c, z$ .

Going deeper

For a student

Contiguous relations give recurrences for orthogonal polynomials that are terminating ${}_2F_1$ series, such as Jacobi and Legendre polynomials. They lead to Gauss's continued fraction for ratios of hypergeometric functions. They also show that the vector $(F, F')$ at shifted parameters is obtained from the original one by a matrix with rational entries. This is the starting point of the theory of hypergeometric difference equations.

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Tag	`TX5AV1JTF5`
URI	`m2://TX5AV1JTF5`
Kind	statement, theorem
Writing	W0 stub
MSC	33C05
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About	Hypergeometric function
Uses	Hypergeometric function, Hypergeometric series