# §14.3 Definitions and Hypergeometric Representations

## §14.3(i) Interval

The following are real-valued solutions of (14.2.2) when , and .

14.3.1

### ¶ Ferrers Function of the Second Kind

14.3.2

Here and elsewhere in this chapter

is Olver’s hypergeometric function (§15.1).

exists for all values of and . is undefined when .

When , (14.3.1) reduces to

equivalently,

When () (14.3.2) is replaced by its limiting value; see Hobson (1931, §132) for details. See also (14.3.12)–(14.3.14) for this case.

## §14.3(ii) Interval

14.3.6

### ¶ Associated Legendre Function of the Second Kind

14.3.7.

As standard solutions of (14.2.2) we take the pair and , where

and

Like , but unlike , is real-valued when , and , and is defined for all values of and . The notation is due to Olver (1997b, pp. 170 and 178).

## §14.3(iii) Alternative Hypergeometric Representations

where

For further hypergeometric representations of and see Erdélyi et al. (1953a, pp. 123–139), Andrews et al. (1999, §3.1), Magnus et al. (1966, pp. 153–163), and §15.8(iii).