# Research Papers On Ordinary Differential Equations

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• Calculating the value of $C^{k\in\{1,\infty\}}$ class of smoothness real-valued function's derivative in point of $\mathbb{R}^+$ in radius of convergence of its Taylor polynomial (or series), applying an analog of Newton's binomial... more

Calculating the value of $C^{k\in\{1,\infty\}}$ class of smoothness real-valued function's derivative in point of $\mathbb{R}^+$ in radius of convergence of its Taylor polynomial (or series), applying an analog of Newton's binomial theorem and $q$-difference operator. $(P,q)$-power difference introduced in section 5. Additionally, by means of Newton's interpolation formula, the discrete analog of Taylor series, interpolation using $q$-difference and $p,q$-power difference is shown.

Keywords: Power function, Monomial, Polynomial, Power series (mathematics), Exponential function, Power (mathematics), Exponentiation, Mathematical Series, Cube (Algebra), Euler number, Perfect cube, Diophantine equations, Finite difference, Derivative, Divided difference, High order finite difference, Ordinary differential equation, Partial differential equation, Partial derivative, Partial difference, High order derivative, Differential calculus, Calculus, Differential equations, Differentiation, Derivatives, Finite differences, Difference Equations, Numerical Differentiation, Finite difference coefficient, Forward Finite Difference, Backward Finite Difference, Central Finite difference, Binomial coefficient, Newton's Binomial Theorem, Pascal’s triangle, Binomial Series, Binomial theorem, Newton's interpolation formula, Binomial expansion, Pascal triangle, Multinomial theorem, Binomial Sum, Mathematics, Math, Maths, Science, Algebra, Number theory, Numerical analysis, Mathematical analysis, Functional analysis, STEM, Numercal methods, Classical Analysis and ODEs, Analysis of PDEs, General Mathematics, Discrete Mathematics, Applied Mathematics, Calculus of variations, q-derivative, Jackson derivative, q-calculus, h-calculus, Quantum calculus, q-difference, Quantum algebra, Qunatum calculus, Hypergeometric series, Hypergeometric function, Time Scale Calculus, Power quantum calculus, Quantum difference, Quantum variatoinal calculus, h-difference, arXiv, Preprint, Open science, arXiv.org, Open access, 0000-0002-6544-8880, Series Expansion, Taylor's theorem, Taylor's formula, Taylor's series, Taylor's polynomial, Analytic function, Series representation, Polynomial expansion, Taylor theorem, Taylor formula, Taylor series, Taylor polynomial, Maclaurin Series, Petro Kolosov, Kolosov Petro, Kolosov, kolosov_p_1, KolosovP, kolosov_petro, petro-kolosov, kolosov-petro, petrokolosov, kolosov.petro, Kolosov_Petro, petro.kolosov.9

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