For fixed complex q with |q| > 1, the q-logarithm Lq is the meromorphic continuation of the series ∑ n>0 z / q −1 , |z| < |q|, into the whole complex plane. IfK is an algebraic number field, one may ask if 1, Lq 1 , Lq c are linearly independent over K for q, c ∈ K× satisfying |q| > 1, c / q, q2, q3, . . .. In 2004, Tachiya showed that this is true in the Subcase K Q, q ∈ Z, c −1, and the prese...

There are a number of results concerning the generation of a collineation group by two of its elements. A. A. Albert and J. Thompson [1] were the first to exhibit two elements generating the little projective group PSL(d, q) of PG(d − 1, q) (for each d and q). According to a theorem of W. M. Kantor and A. Lubotzky [8], “almost every” pair of its elements generates PSL(d, q) as qd→∞ (asymptotica...

and Applied Analysis 3 where g(t, s) is g (t, s) = {{{{{{{{{{{{{{ {{{{{{{{{{{{{{ { (t − s) q−1 − (1/2) (T − s) q−1 Γ (q) + (T − 2t) (T − s) q−2 4Γ (q − 1) + t (T − t) (T − s) q−3 4Γ (q − 2) , s ≤ t, − (T − s) q−1 2Γ (q) + (T − 2t) (T − s) q−2 4Γ (q − 1) + t (T − t) (T − s) q−3 4Γ (q − 2) , t < s. (14) If we let p → 1− in (7), we obtain G T (t, s) 󵄨󵄨󵄨󵄨p→1− = {{{{{{{{{{{{{{{{{{{ {{{{{{{{{{{{{{{{{...

Contents 1. Introduction 1 2. Carlitz-Fermat quotients 2 3. Non-vanishing of Carlitz-Fermat quotients modulo primes 4 1. Introduction. Let q = p s , where p is a prime and s is a positive integer. Let F q be the finite field of q elements, and set A = F q [T ] and k = F q (T). Let τ be the mapping defined by τ (x) = x q , and let kτ denote the twisted polynomial ring. Let C : A → kτ (a → C a) b...

Homcts(π1,ét(C),Q × ` ) ∼= Hom(Pic0(k),Q` ) + (Ẑ Frobc −−−→ Q` ). where c ∈ C(Fq) is a fixed rational point. Our proof will proceed by upgrading this equality to an equivalence of geometric objects. First, we’ll interpretHomcts(π1,ét(C),Q × ` ) in terms of rank one `-adic local systems on C. Similarly, we’ll interpret the datum of Hom(Pic0(k),Q` ) + (Ẑ Frobc −−−→ Q` ) as a “character sheaf” on ...

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Big q-Jacobi polynomials {Pn(·; a, b, c; q)}∞n=0 are classically defined for 0 < a < q −1, 0 < b < q−1 and c < 0. For the family of little q-Jacobi polynomials {pn(·; a, b|q)}∞n=0, classical considerations restrict the parameters imposing 0 < a < q−1 and b < q−1. In this work we extend both families in such a way that wider sets of parameters are allowed, and we establish orthogonality conditio...

In this paper, we investigate the resolvent operator of singular q-Sturm-Liouville problem defined as − ( 1 / q ) D ⁻ ¹ [D y x )] + [r - λ ]y )=0 −(1/q)Dq⁻¹Dqy(x)+r(x)y(x)=λy(x) , with boundary condition 0 c o s β i n = y(0,λ)cosβ+Dq⁻¹y(0,λ)sinβ=0 where ∈ C λ∈C $r$ is a real function on $[0,∞)$, continuous at zero and r L l ∞ r∈Lq,loc¹(0,∞) . We give an integral representation for some properti...

We study the spatial autocorrelation of energy eigenfunctions psi(n)(q) corresponding to classically chaotic systems in the semiclassical regime. Our analysis is based on the Weyl-Wigner formalism for the spectral average C(epsilon)(q(+),q(-),E) of psi(n)(q(+))psi(*)(n)(q(-)), defined as the average over eigenstates within an energy window epsilon centered at E. In this framework C(epsilon) is ...