Many special functions that depend on parameters satisfy a three-term linear recurrence relation
or equivalently the second-order homogeneous linear difference equation
in which is the forward difference operator (§3.6(i)).
Often and can be expanded in series
with . (For the case see the final paragraph of §2.9(ii) with negative.) This situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)). Formal solutions are
where are the roots of the characteristic equation
. The construction fails if , that is, when .
When , there are linearly independent solutions , , such that
If , or if and , then is recessive and is dominant as . As in the case of differential equations (§§2.7(iii), 2.7(iv)) recessive solutions are unique and dominant solutions are not; furthermore, one member of a numerically satisfactory pair has to be recessive. When and neither solution is dominant and both are unique.
For asymptotic expansions in inverse factorial series see Olde Daalhuis (2004a).
, and higher coefficients are determined by formal substitution.
Alternatively, suppose that . Then the indices are the roots of
Provided that is not zero or an integer, (2.9.1) has independent solutions , , of the form
with and higher coefficients given by (2.9.7) (in the present case the coefficients of and are zero).
If , then (2.9.12) applies only in the case . But there is an independent solution
The coefficients and constant are again determined by formal substitution, beginning with when , or with when . (Compare (2.7.6).)
For analogous results for difference equations of the form
in which and are any integers see Wong and Li (1992a).
For asymptotic approximations to solutions of second-order difference equations analogous to the Liouville–Green (WKBJ) approximation for differential equations (§2.7(iii)) see Spigler and Vianello (1992, 1997) and Spigler et al. (1999). Error bounds and applications are included.
For an introduction to, and references for, the general asymptotic theory of linear difference equations of arbitrary order, see Wimp (1984, Appendix B).
For applications of asymptotic methods for difference equations to orthogonal polynomials, see, e.g. Wang and Wong (2012) and Wong (2014). These methods are particularly useful when the weight function associated with the orthogonal polynomials is not unique or not even known; see, e.g. Dai et al. (2014).