New Aspects of Analytic Number Theory

Takao Komatsu (Hirosaki Univ.) Four term leaping recurrence relations Elsner and Komatsu investigated linear three-term recurrence formulae Zn = T (n)Zn−1+U(n)Zn−2 (n ≥ 2) with sequences of integers {T (n)}n≥0 and {U(n)}n≥0, which are ultimately periodic modulo m. If U(n) 6= 0 for all n ≥ 0, then there exists a three-term leaping recurrence relation of the form W1Zrn+i+W2Zr(n−1)+i+ W3Zr(n−2)+i = 0. In the simplest case T (n) = U(n) = 1 for all n, we know that Fn−LrFn−r+(−1)Fn−2r = 0, where Fn and Ln are Fibonacci and Lucas numbers, respectively. We shall observe four-term relations in various directions.