The nth Delannoy number and the nth Schröder number given by D n = n k=0 n k n + k k and S n = n k=0 n k n + k k 1 k + 1 respectively arise naturally from enumerative combinatorics. Let p be an odd prime. We mainly show that p−1 k=1 D k k 2 ≡ 2 −1 p E p−3 (mod p) and p−1 k=1 S k m k ≡ m 2 − 6m + 1 2m 1 − m 2 − 6m + 1 p (mod p), where (−) is the Legendre symbol, E 0 , E 1 , E 2 ,. .. are Euler n...

In this paper, we consider the relationships between the second order linear recurrences and the permanents and determinants of tridiagonal matrices. 1. Introduction The well-known Fibonacci, Lucas and Pell numbers can be generalized as follows: Let A and B be nonzero, relatively prime integers such that D = A 4B 6= 0: De…ne sequences fung and fvng by, for all n 2 (see [10]), un = Aun 1 Bun 2 (...

We compute liftings of the Nichols algebra of a Yetter-Drinfeld module of Cartan type B 2 subject to the small restriction that the diagonal elements of the braiding matrix are primitive nth roots of 1 with odd n = 5. As well, we compute the lift-ings of a Nichols algebra of Cartan type A 2 if the diagonal elements of the braiding matrix are cube roots of 1; this case was not completely covered...

The algebra B of bicomplex numbers is viewed as a complexification the Archimedean f-algebra hyperbolic D. This lattice-theoretic approach allows us to establish new properties so-called D-norms. In particular, we show that D-norms generate same topology in B. We develop D-trigonometric form number which leads geometric interpretation nth roots terms polyhedral tori. use concepts developed, par...

We compute liftings of the Nichols algebra of a Yetter-Drinfeld module of Cartan type B2 subject to the small restriction that the diagonal elements of the braiding matrix are primitive nth roots of 1 with odd n 6= 5. As well, we compute the liftings of a Nichols algebra of Cartan type A2 if the diagonal elements of the braiding matrix are cube roots of 1; this case was not completely covered i...

We review an algebraic method for constructing degenerate eigenvectors of the transfer matrix of the eight-vertex Cyclic Solid-on-Solid lattice model (8V CSOS model), where the degeneracy increases exponentially with respect to the system size. We consider the elliptic quantum group Eτ,η(sl2) at the discrete coupling constants: 2Nη = m1 + im2τ , where N,m1 and m2 are integers. Then we show that...

In this paper we study the class of differential operators T = Pk j=1 QjD j with polynomial coefficients Qj in one complex variable satisfying the condition degQj ≤ j with equality for at least one j. We show that if degQk < k then the root with the largest modulus of the nth degree eigenpolynomial pn of T tends to infinity when n → ∞, as opposed to the case when degQk = k, which we have treate...