Mixed convolutions of the Jacobsthal type

摘要

In this note we consider three classes of the polynomials: {Jn,mr(x)},{jn,ms(x),} and {fn,mr,s(x)}. For these polynomials some interesting relations are found. The Jacobsthal {Jn(x)} and Jacobsthal–Lucas {jn(x)} polynomials are the special cases of these polynomials, which are considered in [A.F. Horadam, P. Filipponi, Derivative sequences of Jacobsthal and Jacobsthal–Lucas polynomials, The Fibonacci Quarterly 35 (4) (1997) 352–358] and [A.F. Horadam, Jacobsthal representation numbers, The Fibonacci Quarterly 34 (1) (1996) 40–54] while the polynomials Jn,m0(x) and jn,m0(x) are studied in [G.B. Djordjević, Derivative sequences of generalized Jacobsthal and Jacobsthal–Lucas polynomials, The Fibonacci Quarterly 38 (2000) 334–38] and [G.B. Djordjević, Generalized Jacobsthal polynomials, The Fibonacci Quarterly, in press]. Also, the numerical arrays {Jn,30(-1)}, {Jn,30(-3/16),} {jn,30(-1)} and {jn,30(-3/16)} are considered.