We derive two identities for multiple basic hyper-geometric series associated with the unitary $U(n+1)$ group. In order to get the two identities, we first present two known $q$-exponential operator identities which were established in our earlier paper. From the two identities and combining them with the two $U(n+1)$ $q$-Chu-Vandermonde summations established by Milne, we arrive at our results. Using the identities obtained in this paper, we give two interesting identities involving binomial...

We compare the growth of the least common multiple of the numbers $u_{a_1},...,u_{a_n}$ and $|u_{a_1}\cdots u_{a_n}|$, where ${\left(u_n\right)}_{n\ge 0}$ is a Lucas sequence and ${\left(a_n\right)}_{n\ge 0}$ is some sequence of positive integers.

We consider $N$, the number of solutions $(x,y,u,v)$ to the equation ${(-1)}^{u}r{a}^x+{(-1)}^{v}s{b}^y=c$ in nonnegative integers $x,y$ and integers $u,v\in \{0,1\}$, for given integers $a\>1$, $b\>1$, $c\>0$, $r\>0$ and $s\>0$. When $gcd(ra,sb)=1$, we show that $N\le 3$ except for a finite number of cases all of which satisfy $max(a,b,r,s,x,y)\<2·{10}^{15}$ for each solution; when $gcd(a,b)\>1$, we show that $N\le 3$ except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving $N=3$ solutions.

Let $p=1+2^{e+1}q$ be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order $2^{e+1}$ (resp. order $d_{\phi }$ dividing q), and let ψₙ be an even character of conductor $p^{n+1}$ and order pⁿ. We put χ = δφψₙ, whose value is contained in $Kₙ=\mathbb{Q}\left({\zeta }_{(p-1)pⁿ}\right)$. It is well known that the Bernoulli number $B_{1,\chi }$ is not zero, which is shown in an analytic way. In the extreme cases $d_{\phi }=1$ and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: $T{r}_{n/1}\left(\xi B_{1,\chi }\right)\ne 0$ for any...

In this paper, we examine a natural question concerning the divisors of the polynomial $x^n-1$: “How often does $x^n-1$ have a divisor of every degree between $1$ and $n$?” In a previous paper, we considered the situation when $x^n-1$ is factored in $\mathbb{Z}\left[x\right]$. In this paper, we replace $\mathbb{Z}\left[x\right]$ with ${𝔽}_p\left[x\right]$, where $p$ is an arbitrary-but-fixed prime. We also consider those $n$ where this condition holds for all $p$.

If the space $𝒬$ of quadratic forms in ${ℝ}^n$ is splitted in a direct sum ${𝒬}_1\oplus ...\oplus {𝒬}_k$ and if $X$ and $Y$ are independent random variables of ${ℝ}^n$, assume that there exist a real number $a$ such that $E\left(X\right|X+Y)=a(X+Y)$ and real distinct numbers $b_1,...,b_k$ such that $E\left(q\left(X\right)\right|X+Y)=b_{i}q(X+Y)$ for any $q$ in ${𝒬}_i.$ We prove that this happens only when $k=2$, when ${ℝ}^n$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.

Let $M$ be a given nonempty set of positive integers and $S$ any set of nonnegative integers. Let $\overline{\delta }\left(S\right)$ denote the upper asymptotic density of $S$. We consider the problem of finding $$\mu \left(M\right):=\underset{S}{sup}\overline{\delta }\left(S\right),$$ where the supremum is taken over all sets $S$ satisfying that for each $a,b\in S$, $a-b\notin M.$ In this paper we discuss the values and bounds of $\mu \left(M\right)$ where $M=\{a,b,a+nb\}$ for all even integers and for all sufficiently large odd integers $n$ with $a<b$ and $gcd(a,b)=1.$

It is proved that real functions on $ℝ$ which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower $C^1$-functions, or of two strongly paraconvex functions) coincide with semismooth functions on $ℝ$ (i.e. those locally Lipschitz functions on $ℝ$ for which $f_{+}^{\text{'}}\left(x\right)={lim}_{t\to x+}{f}_{+}^{\text{'}}\left(t\right)$ and $f_{-}^{\text{'}}\left(x\right)={lim}_{t\to x-}{f}_{-}^{\text{'}}\left(t\right)$ for each $x$). Further, for each modulus $\omega$, we characterize the class $DS{C}_{\omega }$ of functions on $ℝ$ which can be written as $f=g-h$, where $g$ and $h$ are semiconvex with modulus $C\omega$ (for some $C>0$) using a new...

We consider the integral functional $J\left(u\right)={\int }_{\Omega }\left[f\right(\left|Du\right|\left)-u\right]dx$, $u\in W_0^{1,1}\left(\Omega \right)$, where $\Omega \subset {ℝ}^n$, $n\ge 2$, is a nonempty bounded connected open subset of ${ℝ}^n$ with smooth boundary, and $ℝ\ni s\mapsto f\left(\right|s\left|\right)$ is a convex, differentiable function. We prove that if $J$ admits a minimizer in $W_0^{1,1}\left(\Omega \right)$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball.

We consider Weil sums of binomials of the form $W_{F,d}\left(a\right)={\sum }_{x\in F}\psi (x^d-ax)$, where F is a finite field, ψ: F → ℂ is the canonical additive character, $gcd(d,|F^{\times }\left|\right)=1$, and $a\in F^{\times }$. If we fix F and d, and examine the values of $W_{F,d}\left(a\right)$ as a runs through $F^{\times }$, we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo $|F^{\times }|$). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n...

Let f be an arithmetical function. A set S = x₁,..., xₙ of n distinct positive integers is called multiple closed if y ∈ S whenever x|y|lcm(S) for any x ∈ S, where lcm(S) is the least common multiple of all elements in S. We show that for any multiple closed set S and for any divisor chain S (i.e. x₁|...|xₙ), if f is a completely multiplicative function such that (f*μ)(d) is a nonzero integer whenever d|lcm(S), then the matrix $\left(f(x_i,x_i)\right)$ having f evaluated at the greatest common divisor $(x_i,x_i)$ of...

We show that any ${\Sigma }_s$-product of at most $𝔠$-many $L\Sigma (\le \omega )$-spaces has the $L\Sigma (\le \omega )$-property. This result generalizes some known results about $L\Sigma (\le \omega )$-spaces. On the other hand, we prove that every ${\Sigma }_s$-product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every ${\Sigma }_s$-product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

Let $f^j={\sum }_k{a}_{k}f(2^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${ℝ}^n$ having integer-valued components, j∈ℕ and $a_k\in ℂ$. Let $A_{pq}^s$ be either $B_{pq}^s$ or $F_{pq}^s$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${ℝ}^n.$ The aim of the paper is to clarify under what conditions $\parallel f^j|A_{pq}^s\parallel$ is equivalent to $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.

Melham discovered the Fibonacci identity $F_{n+1}{F}_{n+2}{F}_{n+6}-F{³}_{n+3}=(-1)ⁿFₙ$. He then considered the generalized sequence Wₙ where W₀ = a, W₁ = b, and $Wₙ=p{W}_{n-1}+q{W}_{n-2}$ and a, b, p and q are integers and q ≠ 0. Letting e = pab - qa² - b², he proved the following identity: $W_{n+1}{W}_{n+2}{W}_{n+6}-W{³}_{n+3}=e{q}^{n+1}(p³W_{n+2}-q²W_{n+1})$. There are similar differences of products of Fibonacci numbers, like this one discovered by Fairgrieve and Gould: $FₙF_{n+4}{F}_{n+5}-F{³}_{n+3}={(-1)}^{n+1}{F}_{n+6}$. We prove similar identities. For example, a generalization of Fairgrieve and Gould’s identity is $WₙW_{n+4}{W}_{n+5}-W{³}_{n+3}=eqⁿ(p³W_{n+4}-q{W}_{n+5})$.

Consider the $n\times n$ matrix with $(i,j)$’th entry $gcd(i,j)$. Its largest eigenvalue ${\lambda }_n$ and sum of entries $s_n$ satisfy ${\lambda }_n>s_n/n$. Because $s_n$ cannot be expressed algebraically as a function of $n$, we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S. Hong, R. Loewy (2004). We also conjecture that ${\lambda }_n>6{\pi }^{-2}nlogn$ for all $n$. If $n$ is large enough, this follows from F. Balatoni (1969).