# §2.9 Difference Equations

## §2.9(i) Distinct Characteristic Values

Many special functions that depend on parameters satisfy a three-term linear recurrence relation

2.9.1,

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

2.9.4,

where are the roots of the characteristic equation

2.9.5
2.9.6

, and

. The construction fails iff , 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 proofs see Wong and Li (1992b). For error bounds see Zhang et al. (1996). See also Olver (1967b).

For asymptotic expansions in inverse factorial series see Olde Daalhuis (2004a).

## §2.9(ii) Coincident Characteristic Values

When the roots of (2.9.5) are equal we denote them both by . Assume first . Then (2.9.1) has independent solutions , , such that

where

2.9.10,

, and higher coefficients are determined by formal substitution.

Alternatively, suppose that . Then the indices are the roots of

2.9.11

Provided that is not zero or an integer, (2.9.1) has independent solutions , , of the form

2.9.12,

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 proofs and examples, see Wong and Li (1992b). For error bounds see Zhang et al. (1996).

For analogous results for difference equations of the form

2.9.14

in which and are any integers see Wong and Li (1992a).

## §2.9(iii) Other Approximations

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 discussions of turning points, transition points, and uniform asymptotic expansions for solutions of linear difference equations of the second order see Wang and Wong (2003, 2005).

For an introduction to, and references for, the general asymptotic theory of linear difference equations of arbitrary order, see Wimp (1984, Appendix B).