Recurrence relation
Solving general homogeneous linear recurrence relations

Many homogeneous linear recurrence relations may be solved by means of the generalized hypergeometric series. Special cases of these lead to recurrence relations for the orthogonal polynomials, and many special functions. For example, the solution to

is given by

the Bessel function, while

is solved by

the confluent hypergeometric series. Sequences which are the solutions of linear difference equations with polynomial coefficients are called P-recursive. For these specific recurrence equations algorithms are known which find polynomial, rational or hypergeometric solutions.
detectable_format:constrained_response, length_constraints:number_sentences[14]