"Recurrence Equations", An Integrable System.

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  • Hirota Ryogo
    School of Science and Engineering, Waseda University, Tokyo, 169-8555
  • Yahagi Hideyuki
    School of Science and Engineering, Waseda University, Tokyo, 169-8555

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  • "Recurrence Equation", An Integrable System

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We investige discrete equations which return to the initial states after a finite time regardless of the initial values, which we call “recurrence equations”. The recurrence equation exhibits conserved quantities which are expressed by the fundamental symmetric polynomials of solutions, xn+1, xn+2,…, xn+N to the equations of period N. We show that the recurrence equation is integrable by using algebraic entropy and the conserved quantities. We present new recurrence equations of order k (k≥ 2) which have periods k+1 and 2k+2, respectively.

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