§ 2.9. Difference Equations

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

§ 2.9(i). Distinct Characteristic Values

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

2.9.1 ${w(n+2)+f(n)w(n+1)+g(n)w(n)=0},$ $n=0,1,2,\dots$ ,

Defines:: $f(n)$ : function, $g(n)$ : function, $n$ : nonnegative integer and $w(n)$ : solution
Referenced by:: §2.9(ii), §2.9(ii)
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or equivalently the second-order homogeneous linear difference equation

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

Defines:: $f(n)$ : function, $g(n)$ : function, $n$ : nonnegative integer and $w(n)$ : solution
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in which $\Delta$ is the forward difference operator (§Ch.3).

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

2.9.3

$f(n)\sim\sum _{{s=0}}^{{\infty}}\frac{f_{s}}{n^{s}},$

$g(n)\sim\sum _{{s=0}}^{{\infty}}\frac{g_{s}}{n^{s}}$ , $n\to\infty$ ,

Defines:: $f_{s}$ : coefficients and $g_{s}$ : coefficients
Symbols:: $\sim$ : asymptotically equal, $f(n)$ : function, $g(n)$ : function and $n$ : nonnegative integer
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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 $\rho _{j}^{n}n^{{\alpha _{j}}}\sum _{{s=0}}^{{\infty}}\frac{a_{{s,j}}}{n^{s}},$ $j=1,2$ ,

Defines:: $a_{{s,j}}$ : coefficients and $\rho _{j}$ : roots
Symbols:: $n$ : nonnegative integer
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where $\rho _{1},\rho _{2}$ are the roots of the characteristic equation

2.9.5 $\rho^{2}+f_{0}\rho+g_{0}=0,$

Defines:: $f_{s}$ : coefficients and $g_{s}$ : coefficients
Referenced by:: §2.9(ii)
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2.9.6 $\alpha _{j}=(f_{1}\rho _{j}+g_{1})/(f_{0}\rho _{j}+2g_{0}),$

Defines:: $f_{s}$ : coefficients, $g_{s}$ : coefficients and $\rho _{j}$ : roots
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$a_{{0,j}}=1$ , and

2.9.7 $\rho _{j}(f_{0}+2\rho _{j})sa_{{s,j}}=\sum _{{r=1}}^{{s}}\left(\rho _{j}^{2}2^{{r+1}}\binom{\alpha _{j}+r-s}{r+1}+\rho _{j}\sum _{{q=0}}^{{r+1}}\binom{\alpha _{j}+r-s}{r+1-q}f_{q}+g_{{r+1}}\right)a_{{s-r,j}},$

Symbols:: $f_{s}$ : coefficients, $g_{s}$ : coefficients, $a_{{s,j}}$ : coefficients and $\rho _{j}$ : roots
Referenced by:: §2.9(ii)
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$s=1,2,3,\dots$ . The construction fails iff $\rho _{1}=\rho _{2}$ , 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 $w_{j}(n)\sim\rho _{j}^{n}n^{{\alpha _{j}}}\sum _{{s=0}}^{{\infty}}\frac{a_{{s,j}}}{n^{s}},$ $n\to\infty$ .

Defines:: $w_{j}(n)$ : solutions
Symbols:: $\sim$ : asymptotically equal, $n$ : nonnegative integer, $a_{{s,j}}$ : coefficients and $\rho _{j}$ : roots
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If $|\rho _{2}|>|\rho _{1}|$ , or if $|\rho _{2}|=|\rho _{1}|$ and $\realpart{\alpha _{2}}>\realpart{\alpha _{1}}$ , 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 $\realpart{\alpha _{2}}=\realpart{\alpha _{1}}$ 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 (1967).

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

§ 2.9(ii). Coincident Characteristic Values

Referenced by:: §2.9(i)
Permalink:: http://dlmf.nist.gov/2.9.SS2

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

Defines:: $\rho$ : roots and $\kappa$
Symbols:: $\sim$ : asymptotically equal, $c$ : constant and $w_{j}(n)$ : solutions
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where

2.9.10 $\sqrt{g_{0}}\kappa=\sqrt{2f_{0}f_{1}-4g_{1}},$ $4g_{0}\alpha=g_{0}+2g_{1}$ ,

Defines:: $\kappa$
Symbols:: $f_{s}$ : coefficients and $g_{s}$ : coefficients
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$c_{0}=1$ , 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
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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:: $\sim$ : asymptotically equal, $\rho$ : roots and $w_{j}(n)$ : solutions
Referenced by:: §2.9(ii)
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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 $w_{2}(n)\sim\rho^{n}n^{{\alpha _{2}}}\sum _{{\substack{s=0\\ s\neq\alpha _{2}-\alpha _{1}}}}^{{\infty}}\frac{b_{s}}{n^{s}}+cw_{1}(n)\ln n,$ $n\to\infty$ .

Defines:: $b_{s}$ : coefficients and $c$ : constant
Symbols:: $\sim$ : asymptotically equal, $\rho$ : roots and $w_{j}(n)$ : solutions
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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 $w(n+2)+n^{P}f(n)w(n+1)+n^{Q}g(n)w(n)=0,$

Defines:: $P$ : integer and $Q$ : integer
Symbols:: $f(n)$ : function and $g(n)$ : function
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in which $P$ and $Q$ are any integers see Wong and Li (1992a).

§ 2.9(iii). Other Approximations

Permalink:: http://dlmf.nist.gov/2.9.SS3

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).

§ 2.9. Difference Equations

Contents

§ 2.9(i). Distinct Characteristic Values

§ 2.9(ii). Coincident Characteristic Values

§ 2.9(iii). Other Approximations