# §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 ${w(n+2)+f(n)w(n+1)+g(n)w(n)=0},$ $n=0,1,2,\dots$, ⓘ Symbols: $f(n)$: function, $g(n)$: function, $n$: nonnegative integer and $w(n)$: solution Referenced by: §2.9(ii), §2.9(ii) Permalink: http://dlmf.nist.gov/2.9.E1 Encodings: TeX, pMML, png See also: Annotations for 2.9(i), 2.9 and 2

or equivalently the second-order homogeneous linear difference equation

 2.9.2 $\Delta^{2}w(n)+(2+f(n))\Delta w(n)+(1+f(n)+g(n))w(n)=0,$ $n=0,1,2,\dots$, ⓘ Symbols: $\Delta$: forward difference operator, $f(n)$: function, $g(n)$: function, $n$: nonnegative integer and $w(n)$: solution Permalink: http://dlmf.nist.gov/2.9.E2 Encodings: TeX, pMML, png See also: Annotations for 2.9(i), 2.9 and 2

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

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

 2.9.3 $\displaystyle f(n)$ $\displaystyle\sim\sum_{s=0}^{\infty}\frac{f_{s}}{n^{s}},$ $\displaystyle g(n)$ $\displaystyle\sim\sum_{s=0}^{\infty}\frac{g_{s}}{n^{s}}$, $n\to\infty$,

with $g_{0}\neq 0$. (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 $\rho_{j}^{n}n^{\alpha_{j}}\sum_{s=0}^{\infty}\frac{a_{s,j}}{n^{s}},$ $j=1,2$, ⓘ Symbols: $n$: nonnegative integer, $a_{s,j}$: coefficients and $\rho_{j}$: roots Permalink: http://dlmf.nist.gov/2.9.E4 Encodings: TeX, pMML, png See also: Annotations for 2.9(i), 2.9 and 2

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

 2.9.5 $\rho^{2}+f_{0}\rho+g_{0}=0,$ ⓘ Symbols: $f_{s}$: coefficients and $g_{s}$: coefficients Referenced by: §2.9(ii) Permalink: http://dlmf.nist.gov/2.9.E5 Encodings: TeX, pMML, png See also: Annotations for 2.9(i), 2.9 and 2
 2.9.6 $\alpha_{j}=(f_{1}\rho_{j}+g_{1})/(f_{0}\rho_{j}+2g_{0}),$ ⓘ Symbols: $f_{s}$: coefficients, $g_{s}$: coefficients and $\rho_{j}$: roots Permalink: http://dlmf.nist.gov/2.9.E6 Encodings: TeX, pMML, png See also: Annotations for 2.9(i), 2.9 and 2

$a_{0,j}=1$, and

 2.9.7 $\rho_{j}(f_{0}+2\rho_{j})sa_{s,j}=\sum_{r=1}^{s}\left(\rho_{j}^{2}2^{r+1}% \genfrac{(}{)}{0.0pt}{}{\alpha_{j}+r-s}{r+1}+\rho_{j}\sum_{q=0}^{r+1}\genfrac{% (}{)}{0.0pt}{}{\alpha_{j}+r-s}{r+1-q}f_{q}+g_{r+1}\right)a_{s-r,j},$ ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $f_{s}$: coefficients, $g_{s}$: coefficients, $a_{s,j}$: coefficients and $\rho_{j}$: roots Referenced by: §2.9(ii) Permalink: http://dlmf.nist.gov/2.9.E7 Encodings: TeX, pMML, png See also: Annotations for 2.9(i), 2.9 and 2

$s=1,2,3,\dots$. The construction fails if $\rho_{1}=\rho_{2}$, that is, when $f_{0}^{2}=4g_{0}$.

When $f_{0}^{2}\neq 4g_{0}$, there are linearly independent solutions $w_{j}(n)$, $j=1,2$, such that

 2.9.8 $w_{j}(n)\sim\rho_{j}^{n}n^{\alpha_{j}}\sum_{s=0}^{\infty}\frac{a_{s,j}}{n^{s}},$ $n\to\infty$.

If $|\rho_{2}|>|\rho_{1}|$, or if $|\rho_{2}|=|\rho_{1}|$ and $\Re\alpha_{2}>\Re\alpha_{1}$, then $w_{1}(n)$ is recessive and $w_{2}(n)$ is dominant as $n\to\infty$. 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 $\Re\alpha_{2}=\Re\alpha_{1}$ 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).

## §2.9(ii) Coincident Characteristic Values

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

 2.9.9 $w_{j}(n)\sim\rho^{n}\exp\left((-1)^{j}\kappa\sqrt{n}\right)n^{\alpha}\sum_{s=0% }^{\infty}(-1)^{js}\frac{c_{s}}{n^{s/2}},$

where

 2.9.10 $\sqrt{g_{0}}\kappa=\sqrt{2f_{0}f_{1}-4g_{1}},$ $4g_{0}\alpha=g_{0}+2g_{1}$, ⓘ Symbols: $\kappa$, $f_{s}$: coefficients and $g_{s}$: coefficients Permalink: http://dlmf.nist.gov/2.9.E10 Encodings: TeX, pMML, png See also: Annotations for 2.9(ii), 2.9 and 2

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

Alternatively, suppose that $2g_{1}=f_{0}f_{1}$. Then the indices $\alpha_{1},\alpha_{2}$ are the roots of

 2.9.11 $2g_{0}\alpha^{2}-(f_{0}f_{1}+2g_{0})\alpha+2g_{2}-f_{0}f_{2}=0.$ ⓘ Symbols: $f_{s}$: coefficients and $g_{s}$: coefficients Permalink: http://dlmf.nist.gov/2.9.E11 Encodings: TeX, pMML, png See also: Annotations for 2.9(ii), 2.9 and 2

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 $w_{j}(n)\sim\rho^{n}n^{\alpha_{j}}\sum_{s=0}^{\infty}\frac{a_{s,j}}{n^{s}},$ $n\to\infty$, ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\rho$: roots and $w_{j}(n)$: solutions Referenced by: §2.9(ii) Permalink: http://dlmf.nist.gov/2.9.E12 Encodings: TeX, pMML, png See also: Annotations for 2.9(ii), 2.9 and 2

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

 2.9.13 $w_{2}(n)\sim\rho^{n}n^{\alpha_{2}}\sum_{\begin{subarray}{c}s=0\\ s\neq\alpha_{2}-\alpha_{1}\end{subarray}}^{\infty}\frac{b_{s}}{n^{s}}+cw_{1}(n% )\ln n,$ $n\to\infty$.

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 $\alpha_{2}-\alpha_{1}=1,2,3,\dots$. (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 $w(n+2)+n^{P}f(n)w(n+1)+n^{Q}g(n)w(n)=0,$ ⓘ Symbols: $f(n)$: function, $P$: integer, $Q$: integer and $g(n)$: function Permalink: http://dlmf.nist.gov/2.9.E14 Encodings: TeX, pMML, png See also: Annotations for 2.9(ii), 2.9 and 2

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).