This paper examines necessary and sufficient conditions for the function vs (z), Λ(s, v, z)B using coefficients of Touchard polynomials to be in the classes Q(δ, ζ) and K(δ, ζ) of analytic functions. Furthermore, we estimate certain inclusion relations between the classes Rτ (A1,A2) and K(δ, ζ). Finally, we give a necessary and sufficient condition for an integral operator J vs (z) to be in the class K(δ, ζ). Special cases for the classes of starlike and convex functions is also considered.

One of the main motivations of this paper is to construct generating functions for generalization of the Touchard polynomials (or generalization exponential functions) and certain special numbers. Many novel formulas and relations for these polynomials are found by using the Euler derivative operator and functional equations of these functions. Some novel relations among these polynomials, beta polynomials, Bernstein polynomials, related to Binomial distribution from discrete probability distribution classes, are given.

Using the well-known least-squares weighted residual method (LSM) in coupling with various degrees of Touchard polynomials (TPs), found the numerical solutions of Volterra–Fredholm integro-differential equations (VFIDEs) and mixed Volterra–Fredholm integro-differential equations (MIDEs) of the second kind. There exist many approaches that have evaluated the approximate solution of the integro-differential equations (VFIDEs (MIDEs)) like the Adomian decomposition method and modified Adomian decomposition method, Homotopy analysis method, Taylor polynomials, power series expansion and cubic Legendre spline collocation method. In this work, we presented a method based on combining LSM with TPs is an essential component of the suggested approach. By implementing such a method, a system of algebraic equations can be generated that can be solved by employing well-known linear algebraic methods. Several VFIDEs (MIDEs) examples were solved with a comparatively minimal number of reiterations to show the accuracy and effectiveness of the presented approach when comparing the current method with other methods already accessible in the scientific literature, as well as from the approximate solutions of each of these situations, researchers found that there was an apparent agreement with the exact solutions for some examples. The applicability of the proposed method was proven and the convergence analysis was discussed.

This paper introduces an approach for approximating solutions to multi-high-order fractional-differential equations by employing the Caputo fractional derivative, along with initial conditions. The technique is based on standard collocation points and Touchard polynomials. The linear equation and its initial conditions can be transformed into matrix relations by the new method, making it easier to solve a linear algebraic equation with generalized Touchard coefficients as unknowns. Also, emphasizing computational efficiency, the method is illustrated with examples.

Analysis of moment of inertia and magnetic properties under varying magnetic fields via Touchard polynomials collocation technique

Some new operators based on Touchard polynomials

Abstract In this paper, we study the ranges of the Schwartz space $\mathcal {S}$ and its dual $\mathcal {S}'$ (space of tempered distributions) under the Bargmann transform. The characterization of these two ranges leads to interesting reproducing kernel Hilbert spaces whose reproducing kernels can be expressed, respectively, in terms of the Touchard polynomials and the hypergeometric functions. We investigate the main properties of some associated operators and introduce two generalized Bargmann transforms in this framework. This can be considered as a continuation of an interesting research path that Neretin started earlier in his book on Gaussian integral operators.

Since finding solutions to integral equations is usually challenging analytically, approximate methods are often required, one of which is based on Touchard polynomials. This paper examines the necessary constraints for the functions , and the integral operator , defined by Touchard polynomials, to be in the comprehensive subclass ∁ η ( q 3 , q 2 , q 1 , q 0 ) of analytic functions. The originality and potential impact of this research may inspire future investigators to identify new sufficient constraints for functions in the subclass ∁ η ( q 3 , q 2 , q 1 , q 0 ) across various special functions, particularly hypergeometric, Dini, and Sturve functions.

In this investigation, we present a new collection of analytic functions that includes Touchard polynomials. We then aim to calculate the Maclaurin coefficients |𝑎2 | and |𝑎3 | and address the Fekete-Szegö functional problem within this specific subfamily. Additionally, we demonstrate several new outcomes by specifying the parameters used in our main findings.

In this paper we study the polynomial $\D_n(x)=n!\sum_{j=0}^n x^j/j!$, which is a variant of derangement polynomials. First we obtain an asymptotic expansion for $\D_n(x)$ with coefficients in terms of Touchard polynomials. Then, we compute the moments $\sum_{n=0}^\infty(\e^xn!-\D_n(x))^k$ for any integer $k\geqslant 1$ and any real $x\in[0,1)$.

The proposed research is on discrete Erlang mixtures. Properties of the mixed distributions analyzed include raw and central moments, which have been derived in terms of moments of the mixing distributions. Cumulants obtained from the cumulant generating functions were also used in deriving the moments. The posterior distribution and posterior moments are also among properties presented. Bayesian, moments and maximum likelihood methods have been applied in parameter estimation. Additionally, the mixture distributions have been fitted to two data sets to test their goodness of fit. Some methods and special functions used in the study are the exponential series, logarithmic series, geometric series, modified Bessel function of the first kind, and the Touchard polynomials. The discrete mixing distributions used are the geometric, Poisson and logarithmic.

In this paper, we propose quantum mechanical operator formalism for Touchard polynomials whose generating function is [Formula: see text] That is replacing [Formula: see text] by [Formula: see text] where [Formula: see text] is the number operator, and using the method of integration within ordered product we find that [Formula: see text] is just the normal ordering form [Formula: see text]. Then by virtue of the Weyl ordering form of quantum mechanical operator, we also introduce a new special polynomial whose generating function is [Formula: see text] With use of the Weyl ordering form of [Formula: see text] we prove [Formula: see text] where [Formula: see text] denotes Weyl ordering. It seems that quantum mechancal operator formalism presents a new and simpler approach for generalizing Touchard polynomial theory.

A numerical approach to solve Non- linear Fredholm integro- differential (NLFID) equation of the 1st order and 2nd type has been proposed. The approach was based upon Touchard polynomials. The non-linear Fredholm integro- differential equation was changed into a system of non- linear algebraic equations and solved using Newton repeating approach. The proposed approach was evaluated by displaying three numerical problems, and the approximate numerical solutions were compared with exact solution and four methods in the literature. MATLAB R2018b was used to perform all calculations and graphs.

The current paper aims to study bi- starlike functions. It presents a new class using Touchard polynomial, which fulfills the subordination conditions. In addition, an estimation of the Taylor-Maclaurin coefficients |a2| and |a3| was found.

The aim of this work is to discuss some conditions for Touchard polynomials to be in the classes TBb(ρ,σ) and TKb(ρ,σ). Also, we obtain some connection between Rη(D,E) and TKb(ρ,σ). Also, we investigate several mapping properties involving these subclasses. Further, we discuss the geometric properties of an integral operator related to the Touchard polynomial. Additionally, briefly mentioned are specific instances of our primary results. Also, several particular examples are presented.

The idea of this research is to find approximate solution of Linear Volterra Integro-Differential Equation (LVIDE) of the first order and second kind by using different degrees of the Touchard polynomials is presented. Because the simplicity definition applying this method will give high resolution results in low cost and short time, also can be easily differentiated, integrated and converge to an exact solution.The algorithm and example given are to illustrate the solution in this way and compare it with the exact solution.

Entire functions whose coefficients are polynomials having real and negative roots are built from Touchard polynomials. The particular set of polynomials is shown to have purely complex roots, where we show a connection of these polynomials with certain approximations of the Riemann’s zeta function. Also a certain class of Fourier transforms is shown to have only real roots.

Let SH denote the class of functions f = h + g which are harmonic univalent and sense-preserving in the unite disk U = {z : |z| < 1} where h(z) = z +P∞ k=2 akzk, g(z) =∞Pk=1 bkzk (|b1| < 1). In this paper we establish connections between various subclasses of harmonic univalent functions by applying certain integral operator involving the Touchard Polynomials.

In this paper, a novel numerical method has been outlined for finding a novel solution to nonlinear Volterra integral (NLVI) equation of second type, besides, two kernel type of this equation. Touchard polynomials (TPs) and to different degrees were used for this purpose. The function of approximation was obtained to derive the technique for solving this type of integral equations. Three numerical examples were provided to demonstrate the importance of the method used and the accuracy of the extracted results. In some examples, the results were compared with those of another method. The MATLAB R2018b program was used to carry out all calculations and generate all graphics. Keywords: Nonlinear Volterra Integral Equation, Numerical Method, Touchard Polynomials, Exact Function, Numerical Solution

This paper introduced a numerical method for solving generalized Abel integral (GAI) equations and generalized Abel integro differential (GAID) equations. This method is based upon Touchard polynomials (TPs) approximation. The Touchard polynomials were first presented and the resulting Touchard matrices were utilized to transform the generalized Abel integral and integro differential equations into a system of linear algebraic equations. The results of the presented method were obtained through some examples of the first and second types of equations under study. All results for this method have been compared with those of the presented methods in the literature.