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

Jian-Ping Fang

Displaying similar documents to “Generalizations of Milne’s $U (n + 1)$ $q$ -Chu-Vandermonde summation”

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.

Approach of $q$ -Derivative Operators to Terminating $q$ -Series Formulae

Xiaoyuan Wang, Wenchang Chu (2018)

Communications in Mathematics

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The $q$ -derivative operator approach is illustrated by reviewing several typical summation formulae of terminating basic hypergeometric series.

Lucas factoriangular numbers

Bir Kafle, Florian Luca, Alain Togbé (2020)

Mathematica Bohemica

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We show that the only Lucas numbers which are factoriangular are $1$ and $2$ .

On a generalization of the Pell sequence

Jhon J. Bravo, Jose L. Herrera, Florian Luca (2021)

Mathematica Bohemica

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The Pell sequence ${(P_{n})}_{n = 0}^{\infty}$ is the second order linear recurrence defined by $P_{n} = 2 P_{n - 1} + P_{n - 2}$ with initial conditions $P_{0} = 0$ and $P_{1} = 1$ . In this paper, we investigate a generalization of the Pell sequence called the $k$ -generalized Pell sequence which is generated by a recurrence relation of a higher order. We present recurrence relations, the generalized Binet formula and different arithmetic properties for the above family of sequences. Some interesting identities involving the Fibonacci and generalized Pell numbers...

Repdigits in generalized Pell sequences

Jhon J. Bravo, Jose L. Herrera (2020)

Archivum Mathematicum

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For an integer $k \geq 2$ , let ${(n)}_{n}$ be the $k -$ generalized Pell sequence which starts with $0, ..., 0, 1$ ( $k$ terms) and each term afterwards is given by the linear recurrence $n = 2 n - 1 + n - 2 + \dots + n - k$ . In this paper, we find all $k$ -generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence ${(P_{n}^{(2)})}_{n}$ . ...

Some identities involving differences of products of generalized Fibonacci numbers

Curtis Cooper (2015)

Colloquium Mathematicae

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

$L_{p, q}$ spaces

Joseph Kupka

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CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the $L_{p, q}$ spaces................................. 114. Integral representation of bounded linear functionals on $L_{p, q} (B)$ ........ 235. Examples in $L_{p, q}$ theory...................................................................................

$(m, r)$ -central Riordan arrays and their applications

Sheng-Liang Yang, Yan-Xue Xu, Tian-Xiao He (2017)

Czechoslovak Mathematical Journal

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For integers $m > r \geq 0$ , Brietzke (2008) defined the $(m, r)$ -central coefficients of an infinite lower triangular matrix $G = (d, h) = {(d_{n, k})}_{n, k \in ℕ}$ as $d_{m n + r, (m - 1) n + r}$ , with $n = 0, 1, 2, \dots$ , and the $(m, r)$ -central coefficient triangle of $G$ as $G^{(m, r)} = {(d_{m n + r, (m - 1) n + k + r})}_{n, k \in ℕ} .$ It is known that the $(m, r)$ -central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array $G = (d, h)$ with $h (0) = 0$ and $d (0), h^{'} (0) \neq 0$ , we obtain the generating function of its $(m, r)$ -central coefficients and give an explicit representation for the $(m, r)$ -central Riordan array $G^{(m, r)}$ in terms of the Riordan array $G$ ....

Some new Formulas involving $Γ_{q}$ Functions

Thomas Ernst (2007)

Rendiconti del Seminario Matematico della Università di Padova

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On operators from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ or to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$

Christian Samuel (2010)

Colloquium Mathematicae

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We show that every operator from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ is compact when 1 ≤ p,q < s and that every operator from $ℓ_{s}$ to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$ is compact when 1/p + 1/q > 1 + 1/s.

A localization property for $B_{p q}^{s}$ and $F_{p q}^{s}$ spaces

Hans Triebel (1994)

Studia Mathematica

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Let $f^{j} = \sum_{k} a_{k} f (2^{j + 1} x - 2 k)$ , 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 $A_{p q}^{s}$ be either $B_{p q}^{s}$ or $F_{p q}^{s}$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on $ℝ^{n} .$ The aim of the paper is to clarify under what conditions $∥ f^{j} | A_{p q}^{s} ∥$ is equivalent to $2^{j (s - n / p)} (\sum_{k} | a_{k} |^{p})^{1 / p} ∥ f | A_{p q}^{s} ∥$ .

$Σ_{s}$ -products revisited

Reynaldo Rojas-Hernández (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show that any $Σ_{s}$ -product of at most $𝔠$ -many $L Σ (\leq ω)$ -spaces has the $L Σ (\leq ω)$ -property. This result generalizes some known results about $L Σ (\leq ω)$ -spaces. On the other hand, we prove that every $Σ_{s}$ -product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every $Σ_{s}$ -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...

Repdigits in the base $b$ as sums of four balancing numbers

Refik Keskin, Faticko Erduvan (2021)

Mathematica Bohemica

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The sequence of balancing numbers $(B_{n})$ is defined by the recurrence relation $B_{n} = 6 B_{n - 1} - B_{n - 2}$ for $n \geq 2$ with initial conditions $B_{0} = 0$ and $B_{1} = 1 .$ $B_{n}$ is called the $n$ th balancing number. In this paper, we find all repdigits in the base $b,$ which are sums of four balancing numbers. As a result of our theorem, we state that if $B_{n}$ is repdigit in the base $b$ and has at least two digits, then $(n, b) = (2, 5), (3, 6)$ . Namely, $B_{2} = 6 = {(11)}_{5}$ and $B_{3} = 35 = {(55)}_{6} .$

$C^{*}$ -points vs $P$ -points and $P^{♭}$ -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space $X$ , the point $p \in X$ is called a $C^{*}$ -point if every real-valued continuous function on $C ∖ {p}$ can be extended continuously to $p$ . Every point in an extremally disconnected space is a $C^{*}$ -point. A classic example is the space $𝐖^{*} = ω_{1} + 1$ consisting of the countable ordinals together with $ω_{1}$ . The point $ω_{1}$ is known to be a $C^{*}$ -point as well as a $P$ -point. We supply a characterization of $C^{*}$ -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Pell and Pell-Lucas numbers of the form $- 2^{a} - 3^{b} + 5^{c}$

Yunyun Qu, Jiwen Zeng (2020)

Czechoslovak Mathematical Journal

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In this paper, we find all Pell and Pell-Lucas numbers written in the form $- 2^{a} - 3^{b} + 5^{c}$ , in nonnegative integers $a$ , $b$ , $c$ , with $0 \leq max {a, b} \leq c$ .

On the distribution of $(k, r)$ -integers in Piatetski-Shapiro sequences

Teerapat Srichan (2021)

Czechoslovak Mathematical Journal

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A natural number $n$ is said to be a $(k, r)$ -integer if $n = a^{k} b$ , where $k > r > 1$ and $b$ is not divisible by the $r$ th power of any prime. We study the distribution of such $(k, r)$ -integers in the Piatetski-Shapiro sequence ${⌊ n^{c} ⌋}$ with $c > 1$ . As a corollary, we also obtain similar results for semi- $r$ -free integers.

Displaying similar documents to “Generalizations of Milne’s U ( n + 1 ) q -Chu-Vandermonde summation”

Displaying similar documents to “Generalizations of Milne’s $U (n + 1)$ $q$ -Chu-Vandermonde summation”