Burcu Silindir
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Burcu Silindir has a total of 9 co-authors. Here are the top co-authors by number of shared publications:
- Ahmet Yantir
Yaşar Üniversitesi
7 shared publications - Seçil Gergün
Dokuz Eylül Üniversitesi
4 shared publications Ahmet Yantir
Yaşar University, Department of Mathematics, İzmir, Turkey
1 shared publication- Maciej Błaszak
Institute of Spintronics and Quantum Information, Faculty of Physics and Astronomy, Adam Mickiewicz University, Poznań, Poland
1 shared publication - Mieczysław Cichoń
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland
1 shared publication Seçil Gergün
Dokuz Eylül University, Department of Mathematics, İzmir, Turkey
1 shared publication
Latest Publications
- 28 October 2026
- Published by Hacettepe University in Hacettepe Journal of Mathematics and Statistics
- No. Advanced O
Exploring generalized discrete Bessel functions and cylindrical Laplace equation on $\alpha$-time scale
This article introduces a novel approach for discrete Bessel equations and Bessel functions of the first and second kinds on $\alpha$-time scale. This approach not only addresses existing gaps in discrete, $q$-discrete and $(q,h)$-discrete Bessel equations and functions of the second kind but also provides their unification and extension. We introduce $\alpha$-Bessel equation of order $[\nu]\alpha$, construct $\alpha$-Bessel function of order $[\nu]\alpha$ of the first kind, $J_{\nu}^{\alpha}$, and analyze second linearly independent solutions in two distinct cases. We introduce $\alpha$-Bessel function of order $[\nu]\alpha$ of the second kind, namely $\alpha$-Weber's Bessel function, $Y{\nu}^{\alpha}$, as a special linear combination of $J_{\nu}^{\alpha}$ and $\bar{Y}{\nu}^{\alpha}$, where $\bar{Y}{\nu}^{\alpha}$ is $\alpha$-Neumann's Bessel function. The $\alpha$-Bessel function of the second kind recovers the various Bessel functions of the second kind in proper limits. Additionally, we present modified $\alpha$-Bessel equation and functions of the first and second kinds and state a relationship between first kinds of $\alpha$-Bessel and modified $\alpha$-Bessel functions. As an application, we introduce the $\alpha$-analogue of the Laplace equation in cylindrical coordinates and derive its solutions in terms of $\alpha$-analogues of trigonometric, hyperbolic, Bessel, and modified Bessel functions. Finally, graphs of some $\alpha$-Bessel functions are presented and reductions to several discrete Bessel equations and functions are outlined.- 28 October 2026
- Published by Hacettepe University in Hacettepe Journal of Mathematics and Statistics
- No. Advanced O
Exploring generalized discrete Bessel functions and cylindrical Laplace equation on $\alpha$-time scale
This article introduces a novel approach for discrete Bessel equations and Bessel functions of the first and second kinds on $\alpha$-time scale. This approach not only addresses existing gaps in discrete, $q$-discrete and $(q,h)$-discrete Bessel equations and functions of the second kind but also provides their unification and extension. We introduce $\alpha$-Bessel equation of order $[\nu]\alpha$, construct $\alpha$-Bessel function of order $[\nu]\alpha$ of the first kind, $J_{\nu}^{\alpha}$, and analyze second linearly independent solutions in two distinct cases. We introduce $\alpha$-Bessel function of order $[\nu]\alpha$ of the second kind, namely $\alpha$-Weber's Bessel function, $Y{\nu}^{\alpha}$, as a special linear combination of $J_{\nu}^{\alpha}$ and $\bar{Y}{\nu}^{\alpha}$, where $\bar{Y}{\nu}^{\alpha}$ is $\alpha$-Neumann's Bessel function. The $\alpha$-Bessel function of the second kind recovers the various Bessel functions of the second kind in proper limits. Additionally, we present modified $\alpha$-Bessel equation and functions of the first and second kinds and state a relationship between first kinds of $\alpha$-Bessel and modified $\alpha$-Bessel functions. As an application, we introduce the $\alpha$-analogue of the Laplace equation in cylindrical coordinates and derive its solutions in terms of $\alpha$-analogues of trigonometric, hyperbolic, Bessel, and modified Bessel functions. Finally, graphs of some $\alpha$-Bessel functions are presented and reductions to several discrete Bessel equations and functions are outlined.- 16 December 2025
- Published by Hacettepe University in Hacettepe Journal of Mathematics and Statistics
- No. Advanced O
A new approach to hypergeometric functions via $\alpha$-time scale
This manuscript investigates hypergeometric functions on a general discrete time scale $\mathbb{T}_\alpha$. To this end, we introduce the $\alpha$-beta function, establish its fundamental properties, and explore its connection with the $\alpha$-gamma function. In order to achieve our main objective, we formulate the $\alpha$-analogue of the hypergeometric equation and construct its solutions, referred as the $\alpha$-hypergeometric functions. In parallel with the classical case, we illustrate that the $\alpha$-hypergeometric function encompasses many $\alpha$-elementary functions. Furthermore, we derive several key properties of the $\alpha$-hypergeometric function, including the $\alpha$-analogues of the Gauss summation formula, Pfaff transformation, and Euler’s linear identity, which reduce to their classical and $q$-analogous forms.- 16 December 2025
- Published by Hacettepe University in Hacettepe Journal of Mathematics and Statistics
- No. Advanced O
A new approach to hypergeometric functions via $\alpha$-time scale
This manuscript investigates hypergeometric functions on a general discrete time scale $\mathbb{T}_\alpha$. To this end, we introduce the $\alpha$-beta function, establish its fundamental properties, and explore its connection with the $\alpha$-gamma function. In order to achieve our main objective, we formulate the $\alpha$-analogue of the hypergeometric equation and construct its solutions, referred as the $\alpha$-hypergeometric functions. In parallel with the classical case, we illustrate that the $\alpha$-hypergeometric function encompasses many $\alpha$-elementary functions. Furthermore, we derive several key properties of the $\alpha$-hypergeometric function, including the $\alpha$-analogues of the Gauss summation formula, Pfaff transformation, and Euler’s linear identity, which reduce to their classical and $q$-analogous forms.- Open Access
- 1 January 2025
- Published by Walter de Gruyter GmbH in Demonstratio Mathematica
- Vol. 58 (1)
Some special functions and cylindrical diffusion equation on α -time scale
This article is dedicated to present various concepts on -time scale, including power series, Taylor series, binomial series, exponential function, gamma function, and Bessel functions of the first kind. We introduce the -exponential function as a series, examine its absolute and uniform convergence, and establish its additive identity by employing the -Gauss binomial formula. Furthermore, we define the -gamma function and prove -analogue of the Bohr-Mollerup theorem. Specifically, we demonstrate that the -gamma function is the unique logarithmically convex solution of , , where refers to the -number. In addition, we present Euler’s infinite product form and asymptotic behavior of -gamma function. As an application, we propose -analogue of the cylindrical diffusion equation, from which -Bessel and modified -Bessel equations are derived. We explore the solutions of the -cylindrical diffusion equation using the separation of variables technique, revealing analogues of the Bessel and modified Bessel functions of order zero of the first kind. Finally, we illustrate the graphs of the -analogues of exponential and gamma functions and investigate their reductions to discrete and ordinary counterparts.- 1 January 2025
- Published by Walter de Gruyter GmbH in Demonstratio Mathematica
- Vol. 58 (1)
Open AccessSome special functions and cylindrical diffusion equation on α -time scale
This article is dedicated to present various concepts on -time scale, including power series, Taylor series, binomial series, exponential function, gamma function, and Bessel functions of the first kind. We introduce the -exponential function as a series, examine its absolute and uniform convergence, and establish its additive identity by employing the -Gauss binomial formula. Furthermore, we define the -gamma function and prove -analogue of the Bohr-Mollerup theorem. Specifically, we demonstrate that the -gamma function is the unique logarithmically convex solution of , , where refers to the -number. In addition, we present Euler’s infinite product form and asymptotic behavior of -gamma function. As an application, we propose -analogue of the cylindrical diffusion equation, from which -Bessel and modified -Bessel equations are derived. We explore the solutions of the -cylindrical diffusion equation using the separation of variables technique, revealing analogues of the Bessel and modified Bessel functions of order zero of the first kind. Finally, we illustrate the graphs of the -analogues of exponential and gamma functions and investigate their reductions to discrete and ordinary counterparts. - Open Access
Generalized Polynomials and Their Unification and Extension to Discrete Calculus
In this paper, we introduce a comprehensive and expanded framework for generalized calculus and generalized polynomials in discrete calculus. Our focus is on -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 that combines both jump operators in a convex manner. This allows us to generate a time scale , which provides a new approach to discrete calculus. This beneficial approach enables us to define a general symmetric derivative on time scale , which produces various types of discrete derivatives and forms a basis for new discrete calculus. Moreover, we create some polynomials on -time scales using the -operator. These polynomials have similar properties to regular polynomials and expand upon the existing research on discrete polynomials. Additionally, we establish the -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.Open AccessGeneralized Polynomials and Their Unification and Extension to Discrete Calculus
In this paper, we introduce a comprehensive and expanded framework for generalized calculus and generalized polynomials in discrete calculus. Our focus is on -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 that combines both jump operators in a convex manner. This allows us to generate a time scale , which provides a new approach to discrete calculus. This beneficial approach enables us to define a general symmetric derivative on time scale , which produces various types of discrete derivatives and forms a basis for new discrete calculus. Moreover, we create some polynomials on -time scales using the -operator. These polynomials have similar properties to regular polynomials and expand upon the existing research on discrete polynomials. Additionally, we establish the -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. - Open Access
- 30 January 2023
- Published by IntechOpen
Generalized quantum polynomials
On a general time scale, polynomials, Taylors 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 insince 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. - 30 January 2023
- Published by IntechOpen
Open AccessGeneralized quantum polynomials
On a general time scale, polynomials, Taylors 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 insince 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. - Open Access
- 25 January 2023
- Published by Taylor & Francis in Applied Mathematics in Science and Engineering
- Vol. 31 (1)
Power function and binomial series on T ( q , h )
This article is devoted to present -analogue of power function which satisfies additivity and derivative properties similar to the ordinary power function. In the light of nabla -power function, we present -analogue of binomial series and conclude that such power function is -analytic. We prove the analyticity by showing that both the power function and its absolutely convergent Taylor series solve the same IVP. Finally, we present the reductions of -binomial series to classical binomial series, Gauss' binomial and Newton's binomial formulas.- 25 January 2023
- Published by Taylor & Francis in Applied Mathematics in Science and Engineering
- Vol. 31 (1)
Open AccessPower function and binomial series on T ( q , h )
This article is devoted to present -analogue of power function which satisfies additivity and derivative properties similar to the ordinary power function. In the light of nabla -power function, we present -analogue of binomial series and conclude that such power function is -analytic. We prove the analyticity by showing that both the power function and its absolutely convergent Taylor series solve the same IVP. Finally, we present the reductions of -binomial series to classical binomial series, Gauss' binomial and Newton's binomial formulas. - Open Access
- 1 January 2021
- Published by National Library of Serbia in Filomat
- Vol. 35 (11) , 3855-3877
Gauss’s binomial formula and additive property of exponential functions on T(q,h)
In this article, we focus our attention on (q,h)-Gauss?s binomial formula from which we discover the additive property of (q; h)-exponential functions. We state the (q,h)-analogue of Gauss?s binomial formula in terms of proper polynomials on T(q,h) which own essential properties similar to ordinary polynomials. We present (q,h)-Taylor series and analyze the conditions for its convergence. We introduce a new (q,h)-analytic exponential function which admits the additive property. As consequences, we study (q,h)-hyperbolic functions, (q,h)-trigonometric functions and their significant properties such as (q,h)-Pythagorean Theorem and double-angle formulas. Finally, we illustrate our results by a first order (q,h)-difference equation, (q,h)-analogues of dynamic diffusion equation and Burger?s equation. Introducing (q,h)-Hopf-Cole transformation, we obtain (q,h)-shock soliton solutions of Burger?s equation.
- 1 January 2021
- Published by National Library of Serbia in Filomat
- Vol. 35 (11) , 3855-3877
Open AccessGauss’s binomial formula and additive property of exponential functions on T(q,h)
In this article, we focus our attention on (q,h)-Gauss?s binomial formula from which we discover the additive property of (q; h)-exponential functions. We state the (q,h)-analogue of Gauss?s binomial formula in terms of proper polynomials on T(q,h) which own essential properties similar to ordinary polynomials. We present (q,h)-Taylor series and analyze the conditions for its convergence. We introduce a new (q,h)-analytic exponential function which admits the additive property. As consequences, we study (q,h)-hyperbolic functions, (q,h)-trigonometric functions and their significant properties such as (q,h)-Pythagorean Theorem and double-angle formulas. Finally, we illustrate our results by a first order (q,h)-difference equation, (q,h)-analogues of dynamic diffusion equation and Burger?s equation. Introducing (q,h)-Hopf-Cole transformation, we obtain (q,h)-shock soliton solutions of Burger?s equation.
- Open Access
- 1 January 2019
- Published by National Library of Serbia in Filomat
- Vol. 33 (15) , 4907-4922
Generalized quantum exponential function and its applications
This article aims to present (q; h)-analogue of exponential function which unifies, extends hand q-exponential functions in a convenient and efficient form. For this purpose, we introduce generalized quantum binomial which serves as an analogue of an ordinary polynomial. We state (q,h)-analogue of Taylor series and introduce generalized quantum exponential function which is determined by Taylor series in generalized quantum binomial. Furthermore, we prove existence and uniqueness theorem for a first order, linear, homogeneous IVP whose solution produces an infinite product form for generalized quantum exponential function. We conclude that both representations of generalized quantum exponential function are equivalent. We illustrate our results by ordinary and partial difference equations. Finally, we present a generic dynamic wave equation which admits generalized trigonometric, hyperbolic type of solutions and produces various kinds of partial differential/difference equations.
- 1 January 2019
- Published by National Library of Serbia in Filomat
- Vol. 33 (15) , 4907-4922
Open AccessGeneralized quantum exponential function and its applications
This article aims to present (q; h)-analogue of exponential function which unifies, extends hand q-exponential functions in a convenient and efficient form. For this purpose, we introduce generalized quantum binomial which serves as an analogue of an ordinary polynomial. We state (q,h)-analogue of Taylor series and introduce generalized quantum exponential function which is determined by Taylor series in generalized quantum binomial. Furthermore, we prove existence and uniqueness theorem for a first order, linear, homogeneous IVP whose solution produces an infinite product form for generalized quantum exponential function. We conclude that both representations of generalized quantum exponential function are equivalent. We illustrate our results by ordinary and partial difference equations. Finally, we present a generic dynamic wave equation which admits generalized trigonometric, hyperbolic type of solutions and produces various kinds of partial differential/difference equations.
- Open Access
- 1 December 2014
- Published by Elsevier in Annals of Physics
- Vol. 351, 152-165
Flat minimal quantizations of Stäckel systems and quantum separability
- 1 December 2014
- Published by Elsevier in Annals of Physics
- Vol. 351, 152-165
Open AccessFlat minimal quantizations of Stäckel systems and quantum separability