K-Fibonacci sequences and minimal winning quota in Parsimonious game

Parsimonious games are a subset of constant sum homogeneous weighted majority games unequivocally described by their free type representation vector. We show that the minimal winning quota of parsimonious games satisfies a second order, linear, homogeneous, finite difference equation with nonconstant coefficients except for uniform games. We provide the solution of such an equation which may be thought as the generalized version of the polynomial expansion of a proper k-Fibonacci sequence. In addition we show that the minimal winning quota is a symmetric function of the representation vector; exploiting this property it is straightforward to prove that twin Parsimonious games, i.e. a couple of games whose free type representations are each other symmetric, share the same minimal winning quota

Generalized commutative quaternion polynomials of the Fibonacci type

Generalized commutative quaternions is a number system which generalizes elliptic, parabolic and hyperbolic quaternions, bicomplex numbers, complex hyperbolic numbers and hyperbolic complex numbers. In this paper we introduce and study generalized commutative quaternion polynomials of the Fibonacci type

The prime geodesic theorem for PSL2(Z[i])\mathrm{PSL}_{2}(\mathbb{Z}[i]) and spectral exponential sums

We shall ponder the Prime Geodesic Theorem for the Picard manifold $\mathcal{M} = \mathrm{PSL}_{2}(\mathbb{Z}[i]) \backslash \mathfrak{h}^{3}$, which asks about the asymptotic behaviour of a counting function for the closed geodesics on $\mathcal{M}$. Let $E_{\Gamma}(X)$ be the error term arising from counting prime geodesics, we then prove the bound $E_{\Gamma}(X) \ll X^{3/2+\epsilon}$ on average, as well as various versions of pointwise bounds. The second moment bound is the pure counterpart of work of Balog et al. for $\Gamma = \mathrm{PSL}_{2}(\mathbb{Z})$, and the main innovation entails the delicate analysis of sums of Kloosterman sums with an explicit evaluation of oscillatory integrals. Our pointwise bounds concern Weyl-type subconvex bounds for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. Interestingly, we are also able to establish an asymptotic law for the spectral exponential sum in the spectral aspect for a cofinite Kleinian group $\Gamma$. Finally, we produce numerical experiments of its behaviour, visualising that $E_{\Gamma}(X)$ obeys a conjectural bound of the size $O(X^{1+\epsilon})$.Comment: Numerous improvements to the exposition; improved the quality of the main theorem (Theorem 1.1) and achieved additional theorems such as Theorems 1.4, 3.17, 4.1, and 5.

On hyperbolic k-Pell quaternions sequences

In this paper we introduce the hyperbolic k-Pell functions and new classes of quaternions associated with this type of functions are presented. In addition, the Binet formulas, generating functions and some properties of these functions and quaternions sequences are studied. Keywords: Quaternions, Hyperbolic functions, k-Pell sequence, Binet’s identity, Generating functions. MSC: 11B37, 11R52, 05A15, 11B83