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RSolve

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RSolve[eqn, a[n], n]
solves a recurrence equation for a[n].
RSolve[{eqn1, eqn2, ...}, {a1[n], a2[n], ...}, n]
solves a system of recurrence equations.
RSolve[eqn, a[n1, n2, ...], {n1, n2, ...}]
solves a partial recurrence equation.
  • RSolve[eqn, a, n] gives solutions for a as pure functions.
  • The equations can involve objects of the form a[n+Lambda] where Lambda is a constant, or in general, objects of the form a[Psi[n]], a[Psi[Psi[n]], a[Psi[...[Psi[n]]...]], where Psi can have forms such as:
n+Lambdaarithmetic difference equation
Mu ngeometric or q-difference equation
Mu n+Lambdaarithmetic-geometric functional difference equation
Mu nAlphageometric-power functional difference equation
linear fractional functional difference equation
  • Equations such as a[0]Equalval can be given to specify end conditions.
  • If not enough end conditions are specified, RSolve will give general solutions in which undetermined constants are introduced.
  • For partial recurrence equations, RSolve generates arbitrary functions C[n][...].
  • Solutions given by RSolve sometimes include sums that cannot be carried out explicitly by Sum. Dummy variables with local names are used in such sums.
  • RSolve sometimes gives implicit solutions in terms of Solve.
  • RSolve handles both ordinary difference equations and q-difference equations.
  • RSolve handles difference-algebraic equations as well as ordinary difference equations.
  • RSolve can solve linear recurrence equations of any order with constant coefficients. It can also solve many linear equations up to second order with nonconstant coefficients, as well as many nonlinear equations.
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