Congruences related to dual sequences and Catalan numbers.

During the study of dual sequences, Z.-W. Sun introduced the polynomials D n (x , y) = ∑ k = 0 n n k x k y k and S n (x , y) = ∑ k = 0 n n k x k − 1 − x k y k. Many related congruences were also established and conjectured. Here we generalize some of them by determining ∑ k = 0 p − 1 D k (x 1 , y 1) D k (x 2 , y 2) (mod p) and ∑ k = 0 p − 1 S k (x 1 , y 1) S k (x 2 , y 2) (mod p) for odd primes p and p -adic integers x i , y i with i ∈ { 1 , 2 }. In addition, we also characterize ∑ n = 0 p − 1 ∑ k = 0 n n k C k a k 2 (mod p) , where C k denotes the k th Catalan number, a ∈ Z ∖ { 0 } with gcd (a , p) = 1. These results confirm and generalize some of Z.-W. Sun's conjectures. [ABSTRACT FROM AUTHOR]