Power function and binomial series on T
 
 (
 q
 ,
 h
 )

Gergün, Seçil; Silindir, Burcu; Yantir, Ahmet

doi:10.1080/27690911.2023.2168657

Cited by 3 publications

(5 citation statements)

References 17 publications

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“…We showed that such polynomials recover delta q-, delta h-, nabla q-, nabla h-and ordinary polynomials. We emphasize that the study on generalized quantum polynomials not only unify polynomials (and related subjects) on h-lattice sets, quantum numbers and on but also create a paradigm on the theory of special functions (power functions [9], hypergeometric functions, Bernstein polynomials, Bernoulli polynomials, etc.) and combinatorics.…”

Section: Discussionmentioning

confidence: 99%

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Generalized quantum polynomials

Silindir¹,

Yantir²

2023

Recent Research in Polynomials [Working Title]

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On a general time scale, polynomials, Taylor's formula and related subjects are described in terms of implicit and inefficient recursive relations. In this work, our primary goal is to construct proper polynomials, namely delta and nabla generalized quantum polynomials, on (q,h)-time scales explicitly. We show that generalized quantum polynomials play the same roles on (q,h)-time scales as ordinary polynomials play in $\mathbb{R}$ since they obey the additive properties and Leibnitz rules. Such polynomials which recover falling/rising and q-falling/q-rising factorials are constructed by the frame of forward and backward shifts. Additionally, we present delta- and nabla-Gauss' binomial formulas which provide many applications.

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Section: Discussionmentioning

confidence: 99%

“…One of the most significant advantage of q, h ðÞ time scales is to allow us to study q-calculus, h-calculus, and ordinary calculus on one hand. By the definition of jump operators (9), these operators reduce to…”

Section: Preliminariesmentioning

confidence: 99%

Generalized quantum polynomials

Silindir¹,

Yantir²

2023

Recent Research in Polynomials [Working Title]

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“…When the tuning fork is heated, there are changes in the density, length of the tines, thickness of the bar, and young's modulus of the tuning fork which cause changes in equation ( 1) [22], [23]. Using the Newton binomial series, these changes can be written into equation ( 2) [24]- [26].…”

Section: Derivation Of Thetuning Fork Frequency Equationmentioning

confidence: 99%

Experiment to Determine the Relationship Between Temperature and Tuning Fork Frequency

Hasan,

Natalisanto,

Zarkasi

2024

IPR

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A tuning fork is a special tool made of metal and shaped like the letter U with one handle. Tuning forks can produce certain frequencies; usually, the value is written on the handle. This study aims to investigate the relationship between temperature and changes in tuning fork frequency and model it using polynomial regression. This research uses laboratory experiments with tuning forks that have frequencies of 341.3 Hz, 426.5 Hz, and 512 Hz, then the temperature on the tuning fork is varied from 30C to 220C with a difference of 10C. The study's results adjusted R-Square values sequentially 0.94745, 0.99565, and 0.97721, which stated the relationship between temperature and frequency changes. The Adjusted R-Square value close to 1 means that changes in temperature on the tuning fork greatly influence changes in the frequency produced by the tuning fork. The polynomial regression model used is very suitable

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“…The nabla analysis of T x 0 (q,h) when q > 1, x 0 > h 1−q is studied in [23,29,30] and the delta analysis of T x 0 (q,h) when 0 < q < 1, x 0 < h 1−q is studied in [22,31]. In [21], a generalized time scale is introduced as (T, α), depending on an arbitrary operator α : T → T. Motivated by this paper, our primary goal is to unify and extend the (q, h)-time scale (1) using the power of symmetry of introduced concepts.…”

Section: α-Time Scalesmentioning

confidence: 99%

“…The time scale (10) allows us to unify results that are obtained for the time scale T x 0 (q,h) [22,23,[29][30][31] for t ∈ {0, 1} and provides extensions for t ∈ (0, 1). The reductions to T x 0 (q,h) , hZ, K q , and R will be mentioned throughout this paper.…”

Section: α-Time Scalesmentioning

confidence: 99%

Generalized Polynomials and Their Unification and Extension to Discrete Calculus

Cichoń,

Silindir,

Yantir

et al. 2023

Symmetry

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In this paper, we introduce a comprehensive and expanded framework for generalized calculus and generalized polynomials in discrete calculus. Our focus is on (q;h)-time scales. Our proposed approach encompasses both difference and quantum problems, making it highly adoptable. Our framework employs forward and backward jump operators to create a unique approach. We use a weighted jump operator α that combines both jump operators in a convex manner. This allows us to generate a time scale α, which provides a new approach to discrete calculus. This beneficial approach enables us to define a general symmetric derivative on time scale α, which produces various types of discrete derivatives and forms a basis for new discrete calculus. Moreover, we create some polynomials on α-time scales using the α-operator. These polynomials have similar properties to regular polynomials and expand upon the existing research on discrete polynomials. Additionally, we establish the α-version of the Taylor formula. Finally, we discuss related binomial coefficients and their properties in discrete cases. We demonstrate how the symmetrical nature of the derivative definition allows for the incorporation of various concepts and the introduction of fresh ideas to discrete calculus.

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Power function and binomial series on T _{(
q
,
h
)}

Cited by 3 publications

References 17 publications

Generalized quantum polynomials

Generalized quantum polynomials

Experiment to Determine the Relationship Between Temperature and Tuning Fork Frequency

Generalized Polynomials and Their Unification and Extension to Discrete Calculus

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Power function and binomial series on T ( q , h )

Cited by 3 publications

References 17 publications

Generalized quantum polynomials

Generalized quantum polynomials

Experiment to Determine the Relationship Between Temperature and Tuning Fork Frequency

Generalized Polynomials and Their Unification and Extension to Discrete Calculus

Contact Info

Product

Resources

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Power function and binomial series on T _{(
q
,
h
)}