Some Identities Involving Certain Hardy Sums and General Kloosterman Sums

Using the properties of Gauss sums, the orthogonality relation of character sum and the mean value of Dirichlet L-function, we obtain some exact computational formulas for the hybrid mean value involving general Kloosterman sums K ( r , l , λ ; p ) and certain Hardy sums S 1 ( h , q ) ∑ m = 1 p − 1 ∑ s = 1 p − 1 K ( m , n , λ ; p ) K ( s , t , λ ; p ) S 1 ( 2 m s ¯ , p ) , ∑ m = 1 p − 1 ∑ s = 1 p − 1 | K ( m , n , λ ; p ) | 2 | K ( s , t , λ ; p ) | 2 S 1 ( 2 m s ¯ , p ) . Our results not only cover the previous results, but also contain something quite new. Actually the previous authors just consider the case of the principal character λ modulo p, while we consider all the cases.