In this paper, we study sums of finite products of Chebyshev polynomials of the first kind and Lucas polynomials and represent each of them in terms of Chebyshev polynomials of all kinds. Here the coefficients involve terminating hypergeometric functions ₂F₁ and these representations are obtained by explicit computations.

In this paper, we define and investigate modified p-Leonardo, p-Leonardo-Lucas and p-Leonardo sequences as special cases of the generalized Leonardo sequence. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences. Furthermore, we show that there are close relations between modified p-Leonardo, p-Leonardo-Lucas, p-Leonardo numbers and Fibonacci, Lucas numbers.

We evaluate the nested sum $\sum_{a_{n - 1} = c}^{a_n } {\sum_{a_{n - 2} = c}^{a_{n - 1} } { \cdots \sum_{a_0 = c}^{a_1 } {x^{a_0 } } } }$ where $a_n$ and $c$ are any integers and $x$ is a real or complex variable. Consequently, we evaluate multiple sums involving the terms of a general second order sequence, the Horadam sequence (W_j (a, b; p, q)), defined for all non-negative integers j by the recurrence relation W₀ = a, W₁ = b; W_j = pW_j−1 − qW_j−2 (j≥2); where a, b, p and q are arbitrary complex numbers, with p ≠ 0, q ≠ 0.

We evaluate various sums involving the powers of Fibonacci and Lucas numbers.

In this paper, we introduce the generalized Tetranacci quaternions and present some properties of these quaternions.

We evaluate various binomial sums involving the powers of Fibonacci and Lucas numbers.

In this paper, closed forms of the sum formulas $\sum_{k=0}^{n}x^{k}W_{k}^{3}$ and $\sum_{k=1}^{n}x^{k}W_{-k}^{3}$ for the cubes of generalized Fibonacci numbers are presented. As special cases, we give sum formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers.

New approaches to collect in-situ data are needed to complement the high spatial (10~m) and temporal (5-day) resolution of Copernicus Sentinel satellite observations. Making sense of Sentinel observations requires high quality and timely in-situ data for training and validation. Classical ground truth collection is expensive, lacks scale, fails to exploit opportunities for automation, and is prone to sampling error. Here we evaluate the potential contribution of opportunistically exploiting crowd-sourced street-level imagery to collect massive high-quality in-situ data in the context of crop monitoring. This study assesses this potential by answering two questions: 1) what is the spatial availability of these images across the European Union (EU)? and 2) can these images be transformed to useful data? To answer the first question, we evaluated the EU availability of street-level images on Mapillary - the largest open-access platform for such images - against the Land Use and land Cover Area frame Survey (LUCAS) 2018, a systematic surveyed sampling of 337031 points. For 37.78% of the LUCAS points a crowd-sourced image is available within a 2-km buffer, with a mean distance of 816.11 m. We estimate that 9.44% of the EU territory has a crowd-sourced image within 300-m from a LUCAS point, illustrating the huge potential of crowd-sourcing as a complementary sampling tool. After artificial and built up (63.14%), and inland water (43.67%) land cover classes, arable land has the highest availability at 40.78%. To answer the second question, we focus on identifying crops at parcel level using all 13.6 million Mapillary images collected in the Netherlands. Only 1.9% of the contributors generated 75.15% of the images. A procedure was developed to select and harvest the pictures potentially best suited to identify crops using the geometries of 785710 Dutch parcels and the pictures' meta-data such as camera orientation and focal length. Availability of crowd-sourced imagery looking at parcels was assessed for 8 different crop groups with the 2017 parcel level declarations. Parcel revisits during the growing season allowed to track crop growth. Examples illustrate the capacity to recognize crops and their phenological development on crowd-sourced street-level imagery. Consecutive images taken during the same capture track allow selecting the image with the best unobstructed view. In the future, dedicated crop capture tasks can improve image quality and expand coverage in rural areas.

In this paper, we present sums and partial sums of Horadam sequences via generating functions which extends a recent result of Prodinger.

We evaluate some binomial cubic Fibonacci sums.

In this paper, we introduce hybrid numbers with Fibonacci and Lucas hybrid number coefficients. We give the Binet formulas, generating functions, exponential generating functions for these numbers. Then we define an associate matrix for these numbers. In addition, using this matrix, we present two different versions of Cassini identitiy of these numbers.

In this paper, inspired by recent articles of G. Anatriello and G. Vincenzi (see \cite{Ana}), and G. Cerda-Morales (see \cite{Cer1,Cer2}), we will introduce the third-order $\overline{h}$-Jacobsthal and third-order $\overline{h}$-Jacobsthal--Lucas sequences and their associated quaternions. The new results that we have obtained extend most of those obtained in \cite{Cer2}.

In this paper, we introduce the bicomplex Tetranacci and Tetranacci-Lucas quaternions. Moreover, we present Binet's formulas, generating functions, and the summation formulas for those bicomplex quaternions.

We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.

There are many new land use and land cover (LULC) products emerging yet there is still a lack of in-situ data for training, validation, and change detection purposes. The LUCAS (Land Use Cover Area frame Sample) survey is one of the few authoritative in-situ field campaigns, which takes place every three years in European Union member countries. More recently, a study has considered whether citizen science and crowdsourcing could complement LUCAS survey data, e.g., through the FotoQuest Austria mobile app and crowdsourcing campaign. Although the data obtained from the campaign were promising when compared with authoritative LUCAS survey data, there were classes that were not well classified by the citizens, and the photographs submitted through the app were not always of sufficient quality. For this reason, in the latest FotoQuest Go Europe 2018 campaign, several improvements were made to the app to facilitate interaction with the citizens contributing and to improve their accuracy in LULC identification. In addition to extending the locations from Austria to Europe, a change detection component (comparing land cover in 2018 to the 2015 LUCAS photographs) was added, as well as an improved LC decision tree and a near real-time quality assurance system to provide feedback on the distance to the target location, the LULC classes chosen and the quality of the photographs. Another modification was the implementation of a monetary incentive scheme in which users received between 1 to 3 Euros for each successfully completed quest of sufficient quality. The purpose of this paper is to present these new features and to compare the results obtained by the citizens with authoritative LUCAS data from 2018 in terms of LULC and change in LC. We also compared the results between the FotoQuest campaigns in 2015 and 2018 and found a significant improvement in 2018, i.e., a much higher match of LC between FotoQuest Go Europe and LUCAS. Finally, we present the results from a user survey to discuss challenges encountered during the campaign and what further improvements could be made in the future, including better in-app navigation and offline maps, making FotoQuest a model for enabling the collection of large amounts of land cover data at a low cost.

In this paper, closed forms of the summation formulas for generalized Fibonacci numbers are presented. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers.

In this paper, we introduce the generalized dual hyperbolic Fibonacci numbers. As special cases, we deal with dual hyperbolic Fibonacci and dual hyperbolic Lucas numbers. We present Binet's formulas, generating functions and the summation formulas for these numbers. Moreover, we give Catalan's, Cassini's, d'Ocagne's, Gelin-Cesàro's, Melham's identities and present matrices related with these sequences.

In this paper, we present Binet's formulas, generating functions, and the summation formulas for generalized Pentanacci numbers, and as special cases, we investigate Pentanacci and Pentanacci-Lucas numbers with their properties. Also, we define Gaussian generalized Pentanacci numbers and as special cases, we investigate Gaussian Pentanacci and Gaussian Pentanacci-Lucas numbers with their properties. Moreover, we give some identities for these numbers. Furthermore, we present matrix formulations of generalized Pentanacci numbers and Gaussian generalized Pentanacci numbers.

Multi-receiver encryption enables a sender to encrypt a message and transmit the ciphertext to a set of authorized users, while no one out this group of authorized users can decrypt the message. Multi-receiver encryption is of great importance in many sectors such as broadcast communication, cloud computing, wireless communications, networking applications, e-voting, lottery, and medical applications. This paper proposes an efficient multi-receiver public key encryption scheme based on Lucas sequences. The results of the computational analysis show that, projected scheme is better against renown attacks and prevailing anonymous multireceiver algorithms

We present a differential-calculus-based method which allows one to derive more identities from {\it any} given Fibonacci-Lucas identity containing a finite number of terms and having at least one free index. The strength of our method is that no additional information is required about the given original identity. The method readily extends to a generalized Fibonacci sequence.

We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, inverse trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.