Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

Herein, we developed and analyzed a new fractal–fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative. Based on the developed operational matrix and the spectral Tau method, the nonlinear differential problem was reduced to a system of algebraic equations in the unknown expansion coefficients. Accordingly, the resulting system was solved by Newton’s solver with a small initial guess. The efficiency, accuracy, and applicability of the developed numerical method were checked by exhibiting various test problems. The obtained results were also compared with other recent methods, based on the available literature.

Dual addition formulas: the case of continuous q-ultraspherical and q-Hermite polynomials

We settle the dual addition formula for continuous q-ultraspherical polynomials as an expansion in terms of special q-Racah polynomials for which the constant term is given by the linearization formula for the continuous q-ultraspherical polynomials. In a second proof we derive the dual addition formula from the Rahman–Verma addition formula for these polynomials by using the self-duality of the polynomials. We also consider the limit case of continuous q-Hermite polynomials.

Shifted ultraspherical pseudo-Galerkin method for approximating the solutions of some types of ordinary fractional problems

In this work, a technique for finding approximate solutions for ordinary fraction differential equations (OFDEs) of any order has been proposed. The method is a hybrid between Galerkin and collocation methods. Also, this method can be extended to approximate fractional integro-differential equations (FIDEs) and fractional optimal control problems (FOCPs). The spatial approximations with their derivatives are based on shifted ultraspherical polynomials (SUPs). Modified Galerkin spectral method has been used to create direct approximate solutions of linear/nonlinear ordinary fractional differential equations, a system of ordinary fraction differential equations, fractional integro-differential equations, or fractional optimal control problems. The aim is to transform those problems into a system of algebraic equations. That system will be efficiently solved by any solver. Three spaces of collocation nodes have been used through that transformation. Finally, numerical examples show the accuracy and efficiency of the investigated method.

A study of the Fekete-Szegö functional and coefficient estimates for subclasses of analytic functions satisfying a certain subordination condition and associated with the Gegenbauer polynomials

<abstract><p>In this paper, we introduce and study a new subclass of normalized analytic functions, denoted by</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \mathcal F_{\left(\beta,\gamma\right)} \bigg(\alpha,\delta,\mu,H\big(z,C_{n}^{\left(\lambda \right)} \left(t\right)\big)\bigg), $\end{document} </tex-math></disp-formula></p> <p>satisfying the following subordination condition and associated with the Gegenbauer (or ultraspherical) polynomials $ C_{n}^{\left(\lambda\right)}(t) $ of order $ \lambda $ and degree $ n $ in $ t $:</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \alpha \left(\frac{zG^{'}\left(z\right)}{G\left(z\right)} \right)^{\delta}+\left(1-\alpha\right)\left(\frac{zG^{'} \left(z\right)}{G\left(z\right)}\right)^{\mu} \left(1+\frac{zG^{''}\left(z\right)}{G^{'} \left(z\right)} \right)^{1-\mu} \prec H\big(z,C_{n}^{\left(\lambda\right)} \left(t\right)\big), $\end{document} </tex-math></disp-formula></p> <p>where</p> <p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ H\big(z,C_{n}^{\left(\lambda\right)}\left(t\right)\big) = \sum\limits_{n = 0}^{\infty} C_n^{(\lambda)}(t)\;z^n = \left(1-2tz+z^2\right)^{-\lambda}, $\end{document} </tex-math></disp-formula></p> <p><disp-formula> <label/> <tex-math id="FE4"> \begin{document}$ G\left(z\right) = \gamma \beta z^{2} f^{''} \left(z\right)+\left(\gamma-\beta \right)zf^{'} \left(z\right)+\left(1-\gamma+\beta\right)f\left(z\right), $\end{document} </tex-math></disp-formula></p> <p>$ 0\leqq \alpha \leqq 1, $ $ 1\leqq \delta \leqq 2, $ $ 0\leqq \mu \leqq 1, $ $ 0\leqq \beta \leqq \gamma \leqq 1 $, $ \lambda \geqq 0 $ and $ t\in \left(\frac{1}{\sqrt{2}}, 1\right] $. For functions in this function class, we first derive the estimates for the initial Taylor-Maclaurin coefficients $ \left|a_{2}\right| $ and $ \left|a_{3}\right| $ and then examine the Fekete-Szegö functional. Finally, the results obtained are applied to subclasses of normalized analytic functions satisfying the subordination condition and associated with the Legendre and Chebyshev polynomials. The basic or quantum (or $ q $-) calculus and its so-called trivially inconsequential $ (p, q) $-variations have also been considered as one of the concluding remarks.</p></abstract>

New Specific and General Linearization Formulas of Some Classes of Jacobi Polynomials

The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms of the Kampé de Fériet function for some particular choices of the involved parameters. In some cases, the required reduction is performed with the aid of some standard reduction formulas for certain hypergeometric functions of unit argument, while, in other cases, the reduction cannot be done via standard formulas, so we resort to certain symbolic algebraic computation, and specifically the algorithms of Zeilberger, Petkovsek, and van Hoeij. Some new linearization formulas of ultraspherical polynomials and third-and fourth-kinds Chebyshev polynomials are established.

Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series

Several new q-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Zudilin. More concretely, the results in this paper include q-analogues of supercongruences (referring to p-adic identities remaining valid for some higher power of p) established by Long, by Long and Ramakrishna, and several other q-supercongruences. The six basic hypergeometric transformation formulas which are made use of are Watson’s transformation, a quadratic transformation of Rahman, a cubic transformation of Gasper and Rahman, a quartic transformation of Gasper and Rahman, a double series transformation of Ismail, Rahman and Suslov, and a new transformation formula for a nonterminating very-well-poised $${}_{12}\phi _{11}$$ 12 ϕ 11 series. Also, the nonterminating q-Dixon summation formula is used. A special case of the new $${}_{12}\phi _{11}$$ 12 ϕ 11 transformation formula is further utilized to obtain a generalization of Rogers’ linearization formula for the continuous q-ultraspherical polynomials.

A (0;0,2) Interpolation Method to Approximate Functions via Ultraspherical Polynomials

The object of this paper is to demonstrate the existence, explicit characterization and estimation of the polynomial interpolation, related to the weighted (0;0,2) interpolation which satisfies the boundary conditions together with the interpolation conditions at the interval [−1, 1].