Generalized Cochrane sums and Cochrane-Hardy sums

In this paper, the generalized Cochrane sums and Cochrane-Hardy sums are defined. The arithmetic properties of the generalized Cochrane sums are studied, and the Cochrane-Hardy sums are expressed in terms of the generalized Cochrane sums. Analogues of Subrahmanyam's identity and Knopp's theorem are given and proved. Finally, the hybrid mean value of generalized Cochrane sums, Cochrane-Hardy sums and Kloosterman sums is studied, and a few asymptotic formulae are obtained.