Some Identities And Congruences On Ballot Numbers

We first prove the following identity on ballot numbers conjectured by Miana and Romero:For m, n∈Z+, there holds where Cn=1/n+1(2n/n) is the n-th Catalan number. Then we give a summation formula for the following generalized form by applying the Newton interpolation formula. For example, we prove thatWe also give congruences on sums and alternating sums of ballot numbers m (2k+m/k) modulo p. For example, for any prime p>3, we havewhile for any prime p>5, we have...