A certain functional algebra

We study the algebra of functions on the set of natural numbers with respect to the generalized convolution, generated by the generalized translation operator T_nf(k)=f(max (n, k)), n, k. With the help of the generalized Fourier transform, connected with this convolution, we establish numerous identities and recurrence relations, connecting, in particular, sums of powers of natural numbers.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 372–377, March, 1990.