On the $q$-Pell sequences and sums of tails

Alexander E. Patkowski

Displaying similar documents to “On the $q$ -Pell sequences and sums of tails”

On a magnetic characterization of spectral minimal partitions

Bernard Helffer, Thomas Hoffmann-Ostenhof (2013)

Journal of the European Mathematical Society

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Given a bounded open set $Ω$ in $ℝ^{n}$ (or in a Riemannian manifold) and a partition of $Ω$ by $k$ open sets $D_{j}$ , we consider the quantity ${𝚖𝚊𝚡}_{j} λ (D_{j})$ where $λ (D_{j})$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_{j}$ . If we denote by $ℒ_{k} (Ω)$ the infimum over all the $k$ -partitions of ${𝚖𝚊𝚡}_{j} λ (D_{j})$ , a minimal $k$ -partition is then a partition which realizes the infimum. When $k = 2$ , we find the two nodal domains of a second eigenfunction, but the analysis of higher $k$ ’s is non trivial and quite interesting. In this...

On the $k$ -polygonal numbers and the mean value of Dedekind sums

Jing Guo, Xiaoxue Li (2016)

Czechoslovak Mathematical Journal

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For any positive integer $k \geq 3$ , it is easy to prove that the $k$ -polygonal numbers are $a_{n} (k) = (2 n + n (n - 1) (k - 2)) / 2$ . The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet $L$ -functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums $S (a_{n} (k) {\bar{a}}_{m} (k), p)$ for $k$ -polygonal numbers with $1 \leq m, n \leq p - 1$ , and give an interesting computational formula for it.

Subspaces of $L_{p}$ , p > 2, determined by partitions and weights

Dale E. Alspach, Simei Tong (2003)

Studia Mathematica

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Many of the known complemented subspaces of $L_{p}$ have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well known complemented subspaces of $L_{p}$ . It is proved that the class of spaces with such norms is stable under (p,2) sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given...

Rademacher functions in Cesàro type spaces

Sergei V. Astashkin, Lech Maligranda (2010)

Studia Mathematica

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The Rademacher sums are investigated in the Cesàro spaces $C e s_{p}$ (1 ≤ p ≤ ∞) and in the weighted Korenblyum-Kreĭn-Levin spaces $K_{p, w}$ on [0,1]. They span l₂ space in $C e s_{p}$ for any 1 ≤ p < ∞ and in $K_{p, w}$ if and only if the weight w is larger than $t l o g ₂^{p / 2} (2 / t)$ on (0,1). Moreover, the span of the Rademachers is not complemented in $C e s_{p}$ for any 1 ≤ p < ∞ or in $K_{1, w}$ for any quasi-concave weight w. In the case when p > 1 and when w is such that the span of the Rademacher functions is isomorphic to l₂, this span is...

Proof of a conjectured three-valued family of Weil sums of binomials

Daniel J. Katz, Philippe Langevin (2015)

Acta Arithmetica

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We consider Weil sums of binomials of the form $W_{F, d} (a) = \sum_{x \in F} ψ (x^{d} - a x)$ , where F is a finite field, ψ: F → ℂ is the canonical additive character, $g c d (d, | F^{\times} |) = 1$ , and $a \in F^{\times}$ . If we fix F and d, and examine the values of $W_{F, d} (a)$ 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...

Controlling products of currents by higher powers of plurisubharmonic functions

Ahmad K. Al Abdulaali, Hassine El Mir (2020)

Czechoslovak Mathematical Journal

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We discuss the existence of the current $g^{k} T$ , $k \in ℕ$ for positive and closed currents $T$ and unbounded plurisubharmonic functions $g$ . Furthermore, a new type of weighted Lelong number is introduced under the name of weight $k$ Lelong number.

Extension of point-finite partitions of unity

Haruto Ohta, Kaori Yamazaki (2006)

Fundamenta Mathematicae

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A subspace A of a topological space X is said to be $P^{γ}$ -embedded ( $P^{γ}$ (point-finite)-embedded) in X if every (point-finite) partition of unity α on A with |α| ≤ γ extends to a (point-finite) partition of unity on X. The main results are: (Theorem A) A subspace A of X is $P^{γ}$ (point-finite)-embedded in X iff it is $P^{γ}$ -embedded and every countable intersection B of cozero-sets in X with B ∩ A = ∅ can be separated from A by a cozero-set in X. (Theorem B) The product A × [0,1] is $P^{γ}$ (point-finite)-embedded...

Generalized divisor problem for new forms of higher level

Krishnarjun Krishnamoorthy (2022)

Czechoslovak Mathematical Journal

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Suppose that $f$ is a primitive Hecke eigenform or a Mass cusp form for $Γ_{0} (N)$ with normalized eigenvalues $λ_{f} (n)$ and let $X > 1$ be a real number. We consider the sum $𝒮_{k} (X) : = \sum_{n < X} \sum_{n = n_{1}, n_{2}, ..., n_{k}} λ_{f} (n_{1}) λ_{f} (n_{2}) ... λ_{f} (n_{k})$ and show that $𝒮_{k} (X) ≪_{f, ϵ} X^{1 - 3 / (2 (k + 3)) + ϵ}$ for every $k \geq 1$ and $ϵ > 0$ . The same problem was considered for the case $N = 1$ , that is for the full modular group in Lü (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for $k \geq 5$ . Since the result is valid...

Partitioning planar graph of girth 5 into two forests with maximum degree 4

Min Chen, André Raspaud, Weifan Wang, Weiqiang Yu (2024)

Czechoslovak Mathematical Journal

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Given a graph $G = (V, E)$ , if we can partition the vertex set $V$ into two nonempty subsets $V_{1}$ and $V_{2}$ which satisfy $Δ (G [V_{1}]) \leq d_{1}$ and $Δ (G [V_{2}]) \leq d_{2}$ , then we say $G$ has a $(Δ_{d_{1}}, Δ_{d_{2}})$ -partition. And we say $G$ admits an $(F_{d_{1}}, F_{d_{2}})$ -partition if $G [V_{1}]$ and $G [V_{2}]$ are both forests whose maximum degree is at most $d_{1}$ and $d_{2}$ , respectively. We show that every planar graph with girth at least 5 has an $(F_{4}, F_{4})$ -partition.

A note on representation functions with different weights

Zhenhua Qu (2016)

Colloquium Mathematicae

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For any positive integer k and any set A of nonnegative integers, let $r_{1, k} (A, n)$ denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ and $r_{1, l} (A, n) = r_{1, l} (ℕ ∖ A, n)$ hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ for all n ≥ n₀, we have $r_{1, k} (A, n) \to \infty$ as n → ∞.

Integers not of the form $c (2^{a} + 2^{b}) + p^{α}$

Zhi-Wei Sun, Mao-Hua Le (2001)

Acta Arithmetica

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Some weighted norm inequalities for a one-sided version of $g *_{λ}$

L. de Rosa, C. Segovia (2006)

Studia Mathematica

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We study the boundedness of the one-sided operator $g ⁺_{λ, φ}$ between the weighted spaces $L^{p} (M ¯ w)$ and $L^{p} (w)$ for every weight w. If λ = 2/p whenever 1 < p < 2, and in the case p = 1 for λ > 2, we prove the weak type of $g ⁺_{λ, φ}$ . For every λ > 1 and p = 2, or λ > 2/p and 1 < p < 2, the boundedness of this operator is obtained. For p > 2 and λ > 1, we obtain the boundedness of $g ⁺_{λ, φ}$ from $L^{p} ({(M ¯)}^{[p / 2] + 1} w)$ to $L^{p} (w)$ , where ${(M ¯)}^{k}$ denotes the operator M¯ iterated k times.

Generalizations of Milne’s $U (n + 1)$ $q$ -Chu-Vandermonde summation

Jian-Ping Fang (2016)

Czechoslovak Mathematical Journal

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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...

A new proof of the $q$ -Dixon identity

Victor J. W. Guo (2018)

Czechoslovak Mathematical Journal

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We give a new and elementary proof of Jackson’s terminating $q$ -analogue of Dixon’s identity by using recurrences and induction.

On the higher power moments of cusp form coefficients over sums of two squares

Guodong Hua (2022)

Czechoslovak Mathematical Journal

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Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $Γ = SL (2, ℤ)$ . Denote by $λ_{f} (n)$ the $n$ th normalized Fourier coefficient of $f$ . We are interested in the average behaviour of the sum $\sum_{a^{2} + b^{2} \leq x} λ_{f}^{j} (a^{2} + b^{2})$ for $x \geq 1$ , where $a, b \in ℤ$ and $j \geq 9$ is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power $L$ -functions and Rankin-Selberg $L$ -functions.

Displaying similar documents to “On the q -Pell sequences and sums of tails”

Displaying similar documents to “On the $q$ -Pell sequences and sums of tails”