In this section we give asymptotic expansions of PCFs for large values of the parameter that are uniform with respect to the variable , when both and are real. These expansions follow from Olver (1959), where detailed information is also given for complex variables.
With the transformations
With the upper sign in (12.10.2), expansions can be constructed for large in terms of elementary functions that are uniform for (§2.8(ii)). With the lower sign there are turning points at , which need to be excluded from the regions of validity. These cases are treated in §§12.10(ii)–12.10(vi).
Throughout this section the symbol again denotes an arbitrary small positive constant.
uniformly for , where
The coefficients are given by
where and are polynomials in of degree , ( odd), ( even, ). For ,
Higher polynomials can be calculated from the recurrence relation
and the then follow from
uniformly for . The quantities and are defined by
and the coefficients and are given by
uniformly for . Here bars do not denote complex conjugates; instead
and the function has the asymptotic expansion
where and are as in §12.10(ii).
With the same conditions
and the coefficients are the product of and a polynomial in of degree . They satisfy the recursion
starting with . Explicitly,
The modified expansion (12.10.31) shares the property of (12.10.3) that it applies when uniformly with respect to . In addition, it enjoys a double asymptotic property: it holds if either or both and tend to infinity. Observe that if , then , whereas or according as is even or odd. The proof of the double asymptotic property then follows with the aid of error bounds; compare §10.41(iv).
The following expansions hold for large positive real values of , uniformly for . (For complex values of and see Olver (1959).)
The variable is defined by
The function is real for and analytic at . Inversely, with ,
For see (12.10.14). The coefficients and are given by
where is as in §12.10(ii).