Divisibility results on Franel numbers and related polynomials

In this paper, we establish some new divisibility results involving the Franel numbers f(n) = Sigma(n)(k-0) ((n)(k))(3) (n = 0, 1, 2,...) and the polynomials g(n)( x) = Sigma(n)(k-0) ((n)(k))(2) ((2k)(k))(xk) ( n = 0, 1, 2,...). For example, we show that for any positive integer n we have 9/2n(2)(n+1)(2) Sigma(n)(k=1)k(2)(3k+1)(-1)(n-k) f(k) is an element of{1, 2, 3, ...} and 2/n(n+1) Sigma(n)(k=1) k(2)(4k +3)g(k) (2) is an element of {1, 3, 5, ...}, and for any prime p > 3 we have Sigma(p-1)(k=0)k(2) (3k +1)(-1)(k) f(k) = 2/9p(2) (mod p(3)) and Sigma(p-1)(k=0) k(2) (4k +3)g(k)(2) 7/2p (mod p(2)).