scholarly journals How to sum powers of balancing numbers efficiently

2021 ◽  
Vol 27 (1) ◽  
pp. 134-137
Author(s):  
Helmut Prodinger ◽  
Keyword(s):  
Linear Combination ◽  
Fibonacci Numbers ◽  
Partial Sums ◽  
Balancing Numbers ◽  
Binet Formula

Balancing numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of balancing numbers can be summed explicitly. For this, as a first step, a power B_n^l is expressed as a linear combination of B_{mn}.

scholarly journals Summing a Family of Generalized Pell Numbers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Linear Combination ◽  
Computer Algebra ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Partial Sums ◽  
Pell Numbers ◽  
New Family ◽  
Binet Formula

AbstractA new family of generalized Pell numbers was recently introduced and studied by Bród ([2]). These numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power P lnis expressed as a linear combination of Pmn. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter R = 2r, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times.

scholarly journals On Generalization of Fibonacci, Lucas and Mulatu Numbers

2020 ◽  
Vol 1 (3) ◽  
pp. 112-122
Author(s):  
Agung Prabowo
Keyword(s):  
Continued Fraction ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Binet Formula

Fibonacci numbers, Lucas numbers and Mulatu numbers are built in the same method. The three numbers differ in the first term, while the second term is entirely the same. The next terms are the sum of two successive terms. In this article, generalizations of Fibonacci, Lucas and Mulatu (GFLM) numbers are built which are generalizations of the three types of numbers. The Binet formula is then built for the GFLM numbers, and determines the golden ratio, silver ratio and Bronze ratio of the GFLM numbers. This article also presents generalizations of these three types of ratios, called Metallic ratios. In the last part we state the Metallic ratio in the form of continued fraction and nested radicals.

From Fibonacci Numbers to Geometric Sequences and the Binet Formula by Way of the Golden Ratio!

1997 ◽  
Vol 90 (5) ◽  
pp. 386-389
Author(s):  
Angelo S. DiDomenico
Keyword(s):  
Golden Ratio ◽  
Fibonacci Numbers ◽  
School Mathematics ◽  
Evaluation Standards ◽  
Binet Formula

Involving students in open exploration and a search for patterns and relations is a very exciting and rewarding experience. These activities promote stimulating discussions, the application of known skills and relations to unfamiliar settings, and the development of what the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) has called “mathematical power”—the ability to probe and connect and to reason both inductively and deductively. Involvement of this kind can also lead to findings that capture the imagination and that foster a lasting interest in mathematics and an appreciation for its beauty.

scholarly journals Certain Class of Analytic Functions Connected with q -Analogue of the Bessel Function

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Nazek Alessa ◽  
B. Venkateswarlu ◽  
K. Loganathan ◽  
T.S. Karthik ◽  
P. Thirupathi Reddy ◽  
...  
Keyword(s):  
Bessel Function ◽  
Linear Combination ◽  
Analytic Functions ◽  
Partial Sums ◽  
Convex Linear Combination

The focus of this article is the introduction of a new subclass of analytic functions involving q-analogue of the Bessel function and obtained coefficient inequities, growth and distortion properties, radii of close-to-convexity, and starlikeness, as well as convex linear combination. Furthermore, we discussed partial sums, convolution, and neighborhood properties for this defined class.

scholarly journals Logarithmic Spirals and Right-angled Triangles

2020 ◽  
Author(s):  
Rahul Gohil
Keyword(s):  
Linear Combination ◽  
Fibonacci Numbers ◽  
Discrete Form ◽  
Recurrent Functions ◽  
Logarithmic Spirals ◽  
Special Arrangement

This article presents some recurrent functions which we can use to make Right-angledTriangles and through their special arrangement we can obtain logarithmic spirals,and their discrete form. The recurrent functions are obtained through recurrencerelations which can be expressed as a linear combination of fibonacci numbers.

scholarly journals Balancing Numbers and Application

2018 ◽  
Vol 4 ◽  
pp. 137-143
Author(s):  
Ramesh Gautam
Keyword(s):  
Diophantine Equation ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Fibonacci Numbers ◽  
Balancing Numbers

 In this paper, we present about origin of Balancing numbers; It!s connection with Triangular, Pells numbers, and Fibonacci numbers; beginning with connections of balancing numbers with other numbers system, It elaborate the different generating functions of balancing numbers. It also include some amazing recurrence relations; and the application of balancing numbers in solving Diophantine equation.

scholarly journals On a Certain Subclass of Analytic Functions Involving Integral Operator Defined by Polylogarithm Function

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 66
Author(s):  
Danyal Soybaş ◽  
Santosh B. Joshi ◽  
Haridas Pawar
Keyword(s):  
Integral Operator ◽  
Linear Combination ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Distortion Theorem ◽  
Necessary And Sufficient Conditions ◽  
Partial Sums ◽  
Polylogarithm Function ◽  
Necessary And Sufficient

In the present paper, we have introduced a new subclass of analytic functions involving integral operator defined by polylogarithm function. Necessary and sufficient conditions are obtained for this class. Further distortion theorem, linear combination and results on partial sums are investigated.

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