x 1 ( t + 1 ) = ( 1 − m ρ 1 ) x 1 ( t ) + n ρ 2 x 2 ( t ) + h 1 x 2 ( t + 1 ) = ( 1 − m ρ 2 ) x 2 ( t ) + n ρ 1 x 1 ( t ) + h 2Suppose x 1 ( 0 ) and x 2 ( 0 ) are known. How can I find the analytical form of x 1 ( t ) and x 2 ( t )? Without the recursion it is x 1 ( t ) = ( 1 − m ρ 1 ) t x 1 ( 0 ) + 1 − ( 1 − m ρ 1 ) t m ρ 1 h 1 but the recursive form makes it too complicated.

Question

x    1    (  t  +  1  )  =  (  1  −  m      ρ    1    )      x    1    (  t  )  +  n      ρ    2        x    2    (  t  )  +  h  1      x    2    (  t  +  1  )  =  (  1  −  m      ρ    2    )      x    2    (  t  )  +  n      ρ    1        x    1    (  t  )  +  h  2Suppose       x    1    (  0  ) and       x    2    (  0  ) are known. How can I find the analytical form of       x    1    (  t  ) and       x    2    (  t  )? Without the recursion it is      x    1    (  t  )  =  (  1  −  m      ρ    1        )    t        x    1    (  0  )  +            1      −      (      1      −      m              ρ        1                    )        t                    m              ρ        1                  h    1  but the recursive form makes it too complicated.

Sasha Pacheco · Accepted Answer

Define generating functions       X    1    (  t  )  =      ∑          t      ≥      0            x    1    (  t  )      z    t   and       X    2    (  t  )  =      ∑          t      ≥      0            x    2    (  t  )      z    t  , mutiply your recurrences by       z    t   and sum over   t  ≥  0. Then recognize the resulting sums:                                                        X              1                        (            z            )            −                          x              1                        (            0            )                    z                                    =        (        1        −        m                  ρ          1                )                  X          1                (        z        )        +        n                  ρ          2                          X          2                (        z        )        +                              h            1                                1            −            z                                                                                  X              2                        (            z            )            −                          x              2                        (            0            )                    z                                    =        (        1        −        m                  ρ          2                )                  X          1                (        z        )        +        n                  ρ          1                          X          2                (        z        )        +                              h            2                                1            −            z                              Solve this for       X    1    (  z  ) and       X    2    (  z  ), express them as partial fractions and use the generalized binomial theorem to get the coefficients:                (                      −        n            k                  )        =                (                      k        +        n        −        1                    n        −        1                        )        (  −  1      )    k

x 1 </msub> ( t + 1 ) = ( 1 − m ρ<!

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