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Binet's second formula, Hermite's generalization, and two related identities

Open Mathematics, 2023
Legendre was the first to evaluate two well-known integrals involving sines and exponentials. One of these integrals can be used to prove Binet’s second formula for the logarithm of the gamma function.
Boyack Rufus
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The mathematics of generalized Fibonacci sequences: Binet's formula and identities [PDF]

Mathematica Moravica
This article considers a generalized Fibonacci sequence {Vn} with general initial conditions, V0 = a, V1 = b, and a versatile recurrence relation Vn = pVn-1 + qVn-2, where n ≥ 2 and a, b, p and q are any non-zero real numbers. The generating function and
Verma K.L.
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A Training Algorithm for Locally Recurrent Neural Networks Based on the Explicit Gradient of the Loss Function [PDF]

Algorithms
In this paper, a new algorithm for the training of Locally Recurrent Neural Networks (LRNNs) is presented, which aims to reduce computational complexity and at the same time guarantee the stability of the network during the training.
Sara Carcangiu, Augusto Montisci
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On an application of Binet’s second formula

Proceedings of the American Mathematical Society, 2017
In this work we apply the second Binet formula for Euler’s gamma function Γ (
Ruiming Zhang
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Binet’s formula for operator-valued recursive sequences and the operator moment problem

Extracta Mathematicae
We derive a Binet-type formula for operator-valued sequences satisfying linear recurrence relations, extending the classical scalar case to the setting of bounded operators on Hilbert spaces.
A. Ech-charyfy, A. Ghanmi, K. Idrissi, A. Salhi   +3 more
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Some properties of extended remainder of binet’s first formula for logarithm of gamma function [PDF]

Mathematica Slovaca, 2010
Abstract In the paper, we extend Binet’s first formula for the logarithm of the gamma function and investigate some properties, including inequalities, star-shaped and sub-additive properties and the complete monotonicity, of the extended remainder of Binet’s first formula for the logarithm of the gamma function and related functions.
Qui, Feng, Guo, Bai-Ni
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Floor and ceiling functions for Pell numbers [PDF]

Notes on Number Theory and Discrete Mathematics
The analytical study of the Pell number and the role of floor and ceiling functions into their computation is examined. Closed expressions of Pell numbers were initially derived using Binet's formula, followed by an asymptotic behavior study of the ...
İsmail Sulan, Mustafa Aşçı
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New Results on Negative-Indexed Pell Numbers via Matrix Methods

Cumhuriyet Science Journal
In this study, we investigate the Pell and Pell–Lucas numbers sequences and construct matrices whose elements are defined using negative indices of these sequences through Binet’s formula. Identities involving negative-indexed Pell and Pell–Lucas numbers
İbrahim Gökcan, Ümmügülsün Çağlayan, Ali Hikmet Değer   +2 more
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Probabilistic approaches to exploring Binet's type formula for the Tribonacci sequence

AIMS Mathematics
: This paper presents a detailed procedure for deriving a Binet’s type formula for the Tribonacci sequence { T n } . We examine the limiting distribution of a Markov chain that encapsulates the entire sequence { T n } , offering insights into its ...
Skander Hachicha, Najmeddine Attia
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Multiparameter Quantum Cauchy-Binet Formulas [PDF]

Algebras and Representation Theory, 2020
The quantum Cayley-Hamilton theorem for the generator of the reflection equation algebra has been proven by Pyatov and Saponov, with explicit formulas for the coefficients in the Cayley-Hamilton formula. However, these formulas do not give an \emph{easy} way to compute these coefficients.
Karlin, Samuel, Rinott, Yosef
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