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§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 w(n+2)+f(n)w(n+1)+g(n)w(n)=0,

or equivalently the second-order homogeneous linear difference equation

2.9.2 Δ2w(n)+(2+f(n))Δw(n)+(1+f(n)+g(n))w(n)=0,

in which Δ is the forward difference operator (§3.6(i)).

Often f(n) and g(n) can be expanded in series

2.9.3 f(n) s=0fsns,
g(n) s=0gsns,

with g00. (For the case g0=0 see the final paragraph of §2.9(ii) with Q 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 ρjnnαjs=0as,jns,

where ρ1,ρ2 are the roots of the characteristic equation

2.9.5 ρ2+f0ρ+g0=0,
2.9.6 αj=(f1ρj+g1)/(f0ρj+2g0),

a0,j=1, and

2.9.7 ρj(f0+2ρj)sas,j=r=1s(ρj22r+1(αj+r-sr+1)+ρjq=0r+1(αj+r-sr+1-q)fq+gr+1)as-r,j,

s=1,2,3,. The construction fails if ρ1=ρ2, that is, when f02=4g0.

When f024g0, there are linearly independent solutions wj(n), j=1,2, such that

2.9.8 wj(n)ρjnnαjs=0as,jns,

If |ρ2|>|ρ1|, or if |ρ2|=|ρ1| and α2>α1, then w1(n) is recessive and w2(n) is dominant as n. 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 |ρ2|=|ρ1| and α2=α1 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 ρ. Assume first 2g1f0f1. Then (2.9.1) has independent solutions wj(n), j=1,2, such that

2.9.9 wj(n)ρnexp((-1)jκn)nαs=0(-1)jscsns/2,


2.9.10 g0κ=2f0f1-4g1,

c0=1, and higher coefficients are determined by formal substitution.

Alternatively, suppose that 2g1=f0f1. Then the indices α1,α2 are the roots of

2.9.11 2g0α2-(f0f1+2g0)α+2g2-f0f2=0.

Provided that α2-α1 is not zero or an integer, (2.9.1) has independent solutions wj(n), j=1,2, of the form

2.9.12 wj(n)ρnnαjs=0as,jns,

with a0,j=1 and higher coefficients given by (2.9.7) (in the present case the coefficients of as,j and as-1,j are zero).

If α2-α1=0,1,2,, then (2.9.12) applies only in the case j=1. But there is an independent solution

2.9.13 w2(n)ρnnα2s=0sα2-α1bsns+cw1(n)lnn,

The coefficients bs and constant c are again determined by formal substitution, beginning with c=1 when α2-α1=0, or with b0=1 when α2-α1=1,2,3,. (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 w(n+2)+nPf(n)w(n+1)+nQg(n)w(n)=0,

in which P and Q 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).

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).