- §2.9(i) Distinct Characteristic Values
- §2.9(ii) Coincident Characteristic Values
- §2.9(iii) Other Approximations

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

2.9.1 | $$w(n+2)+f(n)w(n+1)+g(n)w(n)=0,$$ | ||

$n=0,1,2,\mathrm{\dots}$, | |||

or equivalently the second-order homogeneous linear difference equation

2.9.2 | $${\mathrm{\Delta}}^{2}w(n)+(2+f(n))\mathrm{\Delta}w(n)+(1+f(n)+g(n))w(n)=0,$$ | ||

$n=0,1,2,\mathrm{\dots}$, | |||

in which $\mathrm{\Delta}$ is the forward difference operator (§3.6(i)).

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

2.9.3 | $f(n)$ | $\sim {\displaystyle \sum _{s=0}^{\mathrm{\infty}}}{\displaystyle \frac{{f}_{s}}{{n}^{s}}},$ | ||

$g(n)$ | $\sim {\displaystyle \sum _{s=0}^{\mathrm{\infty}}}{\displaystyle \frac{{g}_{s}}{{n}^{s}}}$, | |||

$n\to \mathrm{\infty}$, | ||||

with ${g}_{0}\ne 0$. (For the case ${g}_{0}=0$ 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 | $${\rho}_{j}^{n}{n}^{{\alpha}_{j}}\sum _{s=0}^{\mathrm{\infty}}\frac{{a}_{s,j}}{{n}^{s}},$$ | ||

$j=1,2$, | |||

where ${\rho}_{1},{\rho}_{2}$ are the roots of the characteristic equation

2.9.5 | $${\rho}^{2}+{f}_{0}\rho +{g}_{0}=0,$$ | ||

2.9.6 | $${\alpha}_{j}=({f}_{1}{\rho}_{j}+{g}_{1})/({f}_{0}{\rho}_{j}+2{g}_{0}),$$ | ||

${a}_{0,j}=1$, and

2.9.7 | $${\rho}_{j}({f}_{0}+2{\rho}_{j})s{a}_{s,j}=\sum _{r=1}^{s}\left({\rho}_{j}^{2}{2}^{r+1}\left(\genfrac{}{}{0.0pt}{}{{\alpha}_{j}+r-s}{r+1}\right)+{\rho}_{j}\sum _{q=0}^{r+1}\left(\genfrac{}{}{0.0pt}{}{{\alpha}_{j}+r-s}{r+1-q}\right){f}_{q}+{g}_{r+1}\right){a}_{s-r,j},$$ | ||

$s=1,2,3,\mathrm{\dots}$. The construction fails if ${\rho}_{1}={\rho}_{2}$, that is, when ${f}_{0}^{2}=4{g}_{0}$.

When ${f}_{0}^{2}\ne 4{g}_{0}$, there are linearly independent solutions ${w}_{j}(n)$, $j=1,2$, such that

2.9.8 | $${w}_{j}(n)\sim {\rho}_{j}^{n}{n}^{{\alpha}_{j}}\sum _{s=0}^{\mathrm{\infty}}\frac{{a}_{s,j}}{{n}^{s}},$$ | ||

$n\to \mathrm{\infty}$. | |||

If $|{\rho}_{2}|>|{\rho}_{1}|$, or if $|{\rho}_{2}|=|{\rho}_{1}|$ and $\mathrm{\Re}{\alpha}_{2}>\mathrm{\Re}{\alpha}_{1}$, then ${w}_{1}(n)$ is recessive and ${w}_{2}(n)$ is dominant as $n\to \mathrm{\infty}$. 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 $|{\rho}_{2}|=|{\rho}_{1}|$ and $\mathrm{\Re}{\alpha}_{2}=\mathrm{\Re}{\alpha}_{1}$ 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).

When the roots of (2.9.5) are equal we denote them both by $\rho $. Assume first $2{g}_{1}\ne {f}_{0}{f}_{1}$. Then (2.9.1) has independent solutions ${w}_{j}(n)$, $j=1,2$, such that

2.9.9 | $${w}_{j}(n)\sim {\rho}^{n}\mathrm{exp}\left({(-1)}^{j}\kappa \sqrt{n}\right){n}^{\alpha}\sum _{s=0}^{\mathrm{\infty}}{(-1)}^{js}\frac{{c}_{s}}{{n}^{s/2}},$$ | ||

where

2.9.10 | $$\sqrt{{g}_{0}}\kappa =\sqrt{2{f}_{0}{f}_{1}-4{g}_{1}},$$ | ||

$4{g}_{0}\alpha ={g}_{0}+2{g}_{1}$, | |||

${c}_{0}=1$, and higher coefficients are determined by formal substitution.

Alternatively, suppose that $2{g}_{1}={f}_{0}{f}_{1}$. Then the indices ${\alpha}_{1},{\alpha}_{2}$ are the roots of

2.9.11 | $$2{g}_{0}{\alpha}^{2}-({f}_{0}{f}_{1}+2{g}_{0})\alpha +2{g}_{2}-{f}_{0}{f}_{2}=0.$$ | ||

Provided that ${\alpha}_{2}-{\alpha}_{1}$ is not zero or an integer, (2.9.1) has independent solutions ${w}_{j}(n)$, $j=1,2$, of the form

2.9.12 | $${w}_{j}(n)\sim {\rho}^{n}{n}^{{\alpha}_{j}}\sum _{s=0}^{\mathrm{\infty}}\frac{{a}_{s,j}}{{n}^{s}},$$ | ||

$n\to \mathrm{\infty}$, | |||

with ${a}_{0,j}=1$ and higher coefficients given by (2.9.7) (in the present case the coefficients of ${a}_{s,j}$ and ${a}_{s-1,j}$ are zero).

If ${\alpha}_{2}-{\alpha}_{1}=0,1,2,\mathrm{\dots}$, then (2.9.12) applies only in the case $j=1$. But there is an independent solution

2.9.13 | $${w}_{2}(n)\sim {\rho}^{n}{n}^{{\alpha}_{2}}\sum _{\begin{array}{c}s=0\\ s\ne {\alpha}_{2}-{\alpha}_{1}\end{array}}^{\mathrm{\infty}}\frac{{b}_{s}}{{n}^{s}}+c{w}_{1}(n)\mathrm{ln}n,$$ | ||

$n\to \mathrm{\infty}$. | |||

The coefficients ${b}_{s}$ and constant $c$ are again determined by formal substitution, beginning with $c=1$ when ${\alpha}_{2}-{\alpha}_{1}=0$, or with ${b}_{0}=1$ when ${\alpha}_{2}-{\alpha}_{1}=1,2,3,\mathrm{\dots}$. (Compare (2.7.6).)

For analogous results for difference equations of the form

2.9.14 | $$w(n+2)+{n}^{P}f(n)w(n+1)+{n}^{Q}g(n)w(n)=0,$$ | ||

in which $P$ and $Q$ 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 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).