Math 2.0

Difference and functional equations Lemma

General solution of a recurrence with a repeated root

Builds on Shift operatorCharacteristic polynomial of a linear recurrence

Statement

Let AA be the shift operator on sequences, so that (Af)(n)=f(n+1)(Af)(n) = f(n+1). Let rr be a constant and let k≥1k \ge 1 be an integer.

Consider the recurrence (A−r)kf=0(A - r)^k f = 0. Its characteristic polynomial is (x−r)k(x - r)^k, so rr is its only root, with multiplicity kk.

Then the general solution is f(n)=c1rn+c2nrn+c3n2rn+⋯+cknk−1rn,f(n) = c_1 r^n + c_2 n r^n + c_3 n^2 r^n + \dots + c_k n^{k-1} r^n, where c1,…,ckc_1, \dots, c_k are arbitrary constants. That is, each sequence of this form solves the recurrence, and each solution has this form for some constants.

Remarks

For k=1k = 1 the recurrence is f(n+1)=rf(n)f(n+1) = r f(n), and the solutions are c1rnc_1 r^n.

For k=2k = 2 the solutions are (c1+c2n)rn(c_1 + c_2 n) r^n.

The shape matches the differential equation case, where a root of multiplicity mm gives xjerxx^j e^{rx} for 0≤j<m0 \le j < m.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptRecurrence relation, definition, no evidence levelRecurrence relationdefinitionSequence, definition, no evidence levelSequencedefinitionSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionLinear recurrence with constant coefficients, definition, no evidence levelLinear recurrence wit…definitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCharacteristic polynomial of a linear recurrence, definition, no evidence levelCharacteristic polyno…definitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionShift operator, definition, no evidence levelShift operatordefinitionGeneral solution of a recurrence with a repeated root, statement, no evidence levelGeneral solution of a…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Recurrence relation definition
  4. Sequence definition
  5. Set definition
  6. Implication definition
  7. Predicate definition
  8. Linear recurrence with constant coefficients definition
  9. Ordered pair definition
  10. Existential quantifier definition
  11. Universal quantifier definition
  12. Characteristic polynomial of a linear recurrence definition
  13. Cartesian product definition
  14. Subset definition
  15. Uniqueness quantifier definition
  16. Binary relation definition
  17. Function definition
  18. Shift operator definition
  19. General solution of a recurrence with a repeated root statement

Further reading

  1. Web https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_recurrence_rigorous.ptx#L152, source/s_recurrence_rigorous.ptx:152
About this page
TagAGXA4Q7E5J
URIm2://AGXA4Q7E5J
Kindstatement, lemma
WritingW0 stub
MSC39A06
StatusThis copy of the site has no build data. Run m2 status AGXA4Q7E5J.
UsesShift operator, Characteristic polynomial of a linear recurrence
Changed4 November 2026

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