Solves first order, linear recurrence equations.

A first order linear recurrence equation in y(k) is $A\left(k\right)y\left(k+1\right)+B\left(k\right)y\left(k\right)=C\left(k\right)$ where A(k), B(k), and C(k) are independent of y(k).

If A(k) is undefined for some k, then a set of equations may be returned, giving values of y(k) for specific k as well as the general formula.

See the help page for LREtools[REcreate] for the definition of the format of a problem.

$\mathrm{with}\left(\mathrm{LREtools}\right)\:$

$\mathrm{prob}≔\mathrm{REcreate}\left(\left\{y\left(k+1\right)-by\left(k\right)=a\right\}\,\left\{y\left(k\right)\right\}\,\left\{y\left(0\right)=c-\frac{a}{b-1}\right\}\right)$

$\mathrm{prob}≔\mathrm{RESol}\left(\left\{y\left(k+1\right)-by\left(k\right)=a\right\}\,\left\{y\left(k\right)\right\}\,\left\{y\left(0\right)=-\frac{-cb+a+c}{b-1}\right\}\,\mathrm{INFO}\right)$

$\mathrm{firstlin}\left(\mathrm{prob}\right)$

$\frac{b^k\left(-a{\left(\frac{1}{b}\right)}^k+bc-c\right)}{b-1}$

$\mathrm{firstlin}\left(\left\{y\left(k+3\right)-\exp \left(2k+4\right)y\left(k+2\right)=0\right\}\,\left\{y\left(k\right)\right\}\,\varnothing \right)$

${\ⅇ}^{k\left(k-1\right)}y\left(0\right)$

$\mathrm{firstlin}\left(u\left(n+1\right)=\frac{2u\left(n\right)\left(n+1\right)}{n}\,u\left(n\right)\,\varnothing \right)$

$\left\{u\left(0\right)=0\,u\left(n\right)=n{2}^{n-1}u\left(1\right)\right\}$