Fractional Zernike functions

A number of Mittag–Leffler functions are defined in the literature which have many applications across various areas of physical, biological, and applied sciences and are used in solving problems of fractional order differential, integral, and difference equations. This paper aims to define the m‐parameter Mittag–Leffler function, which can be reduced to various already known extensions of the Mittag–Leffler function. Important properties like recurrence relations, differentiation formulae, and integral representations are discussed for the newly defined m‐parameter Mittag–Leffler function. Moreover, the new m‐parameter Mittag–Leffler function is expressed in terms of some well‐known special functions such as generalized hypergeometric function, Mellin‐Barnes integral, Wright generalized hypergeometric function, and Fox H‐function. The paper also deals with various integral transforms like Euler‐Beta, Whittaker, Laplace, and Mellin transforms of the m‐parameter Mittag–Leffler function. Fractional differential and integral operators are considered to highlight certain intriguing properties of the m‐parameter Mittag–Leffler function. We also use the above function to define a generalization of Prabhakar integral and discuss its properties. Further, the relation of the newly defined m‐parameter Mittag–Leffler function with various other functions such as exponential, trigonometric, hypergeometric, and algebraic functions is obtained and represented graphically using MATHEMATICA 12.

In this paper, we give a new generalization of extended beta functions by using generalized Mittag Leffler functions. We investigate its properties and its integral representations. In addition, we establish the generalization of extended hypergeometric and Confluent hypergeometric functions by using the newly extended beta function. Some properties of these extended and Confluent hypergeometric functions such as integral representations, Mellin transformations, differentiation formulas, transformation and summation formulas are also investigated. In this paper, we give a new generalization of extended beta functions by using generalized Mittag Leffler functions. We investigate its properties and its integral representations. In addition, we establish the generalization of extended hypergeometric and Confluent hypergeometric functions by using the newly extended beta function. Some properties of these extended and Confluent hypergeometric functions such as integral representations, Mellin transformations, differentiation formulas, transformation and summation formulas are also investigated. In this paper, we give a new generalization of extended beta functions by using generalized Mittag Leffler functions. We investigate its properties and its integral representations. In addition, we establish the generalization of extended hypergeometric and Confluent hypergeometric functions by using the newly extended beta function. Some properties of these extended and Confluent hypergeometric functions such as integral representations, Mellin transformations, differentiation formulas, transformation and summation formulas are also investigated.

Our goal in this article is to use ideas from symmetric q-calculus operator theory in the study of meromorphic functions on the punctured unit disc and to propose a novel symmetric q-difference operator for these functions. A few additional classes of meromorphic functions are then defined in light of this new symmetric q-difference operator. We prove many useful conclusions regarding these newly constructed classes of meromorphic functions, such as convolution, subordination features, integral representations, and necessary conditions. The technique presented in this article may be used to produce a wide variety of new types of generalized symmetric q-difference operators, which can subsequently be used to investigate a wide variety of new classes of analytic and meromorphic functions related to symmetric quantum calculus.

This review is dedicated to Jacobi polynomials which are a generalization of some well-known classical polynomials such as Gegenbauer polynomials, Legendre polynomials, first and second kinds Chebyshev polynomials and Zernike polynomials. To save effort and time, it is advantageous and sufficient to investigate the properties of Jacobi polynomials rather than considering their special cases separately. For demonstration purposes, we show how to reduce most of the obtained properties of Jacobi polynomials to the corresponding properties of Legendre polynomials such as the generating function, Rodrigues formula, special values and the orthogonality property. Most of the properties of Jacobi polynomials are obtained through their hypergeometric representations such as differential recurrence relations, generating functions, some special values (exact and asymptotic) and some integral expansions of positive integrand. The standard orthogonality property of Jacobi polynomials for the values of the parameters is discussed and some applications of such important property are pointed out. The orthogonality property of Jacobi polynomials considerably affects the positions of their zeros. These zeros are essential in any type of numerical quadrature such as Legendre-Gaussian quadrature, first kind Chebyshev-Gaussian quadrature and second kind Chebyshev-Gaussian quadrature. Moreover the recurrence relations of Jacobi polynomials play an important role in computing the zeros of Jacobi polynomials which are needed in the Gaussian-Jacobi quadrature. A narrative review is provided on some attempts were done in extending the orthogonality property of Jacobi polynomials to non-standard values of the indexes by amending some of the orthogonality conditions such as Sobolev and non-hermitian orthogonality. Integral expansions of Jacobi polynomials with a prominent feature (positive integrand) were presented in terms of other Jacobi polynomials. Such integral expansions should allow a variety of applications for Jacobi polynomials. This review is dedicated to Jacobi polynomials which are a generalization of some well-known classical polynomials such as Gegenbauer polynomials, Legendre polynomials, first and second kinds Chebyshev polynomials and Zernike polynomials. To save effort and time, it is advantageous and sufficient to investigate the properties of Jacobi polynomials rather than considering their special cases separately. For demonstration purposes, we show how to reduce most of the obtained properties of Jacobi polynomials to the corresponding properties of Legendre polynomials such as the generating function, Rodrigues formula, special values and the orthogonality property. Most of the properties of Jacobi polynomials are obtained through their hypergeometric representations such as differential recurrence relations, generating functions, some special values (exact and asymptotic) and some integral expansions of positive integrand. The standard orthogonality property of Jacobi polynomials for the values of the parameters is discussed and some applications of such important property are pointed out. The orthogonality property of Jacobi polynomials considerably affects the positions of their zeros. These zeros are essential in any type of numerical quadrature such as Legendre-Gaussian quadrature, first kind Chebyshev-Gaussian quadrature and second kind Chebyshev-Gaussian quadrature. Moreover the recurrence relations of Jacobi polynomials play an important role in computing the zeros of Jacobi polynomials which are needed in the Gaussian-Jacobi quadrature. A narrative review is provided on some attempts were done in extending the orthogonality property of Jacobi polynomials to non-standard values of the indexes by amending some of the orthogonality conditions such as Sobolev and non-hermitian orthogonality. Integral expansions of Jacobi polynomials with a prominent feature (positive integrand) were presented in terms of other Jacobi polynomials. Such integral expansions should allow a variety of applications for Jacobi polynomials.

In the theory of differential equation and probability, Probabilistic Hermite polynomials Hr(x) = {r=0,1,2,…,n} are the polynomials obtained from derivatives of the standard normal probability density function (pdf) of the form α(x)=1/√2π e^(-1/2 x^2 ). These polynomials played an important role in the Gram-Charlier series expansion of type A and the Edgeworth’s form of the type A series (see [18]). In this paper, we obtained new Probabilistic Hermite polynomials by considering a standard normal distribution with probability density function (pdf) given as β(x)=1/(2√π) e^(-1/4 x^2 ). The generating function, recurrence relations and orthogonality properties are studied. Finally, a differential equation governing these polynomials was presented which enables us to obtain the expression of the polynomial in a closed form.

In this study, we present and explore extended beta matrix functions (EBMFs) and their key properties. By utilizing the beta matrix function (BMF), we introduce novel extensions of the Gauss hypergeometric matrix function (GHMF) and Kummer hypergeometric matrix function (KHMF). We delve into their integral representations, recurrence relations, transformation properties, and differential formulas. Additionally, we investigate their statistical applications, mainly focusing on the beta distribution, and derive expressions for the mean, variance, and moment-generating functions. Furthermore, we apply EBMFs to develop the Appell matrix function (AMF) and Lauricella matrix function (LMF) and their integral forms.

In this paper, we present p+q+m and p+q+2m parametric S-functions, exploring the relationships between these two types of functions. We derive their fundamental properties, including generating functions, recurrence relations, differential formulas and integral representations. From these results, we establish several notablecorollaries. Furthermore, we present a distribution function involving the S-function and compute the Laplace transform of its density function. Additionally, we explore the connections between the S-function and other significant transcendental functions.

In this paper, by the extension of beta functions containing three extra parameters, we generalize Appell's and Lauricella's hypergeometric functions. Some integral representations, transformation formulae, differentiation formulae and recurrence relations are obtained for these new generalizations.

In this chapter we present the basic properties of the classical Mittag-Leffler function \(E_\alpha (z)\) (see (1.0.1)). The material can be formally divided into two parts. Starting from the basic definition of the Mittag-Leffler function in terms of a power series, we discover that for parameter \(\alpha \) with positive real part the function \(E_\alpha (z)\) is an entire function of the complex variable z. Therefore we discuss in the first part the (analytic) properties of the Mittag-Leffler function as an entire function. Namely, we calculate its order and type, present a number of formulas relating it to elementary and special functions as well as recurrence relations and differential formulas, introduce some useful integral representations and discuss the asymptotics and distribution of zeros of the classical Mittag-Leffler function.

This study introduces a new generalization of the Mittag-Leffler function, the k-generalized Mittag-Leffler type function , , , ( ), γ δ α β E z k and investigates its interesting and important basic properties, such as useful relationships between Mittag-Leffler functions, recurrence relations, differential formulas, integral representations, and images of this function under the Euler (Beta), Laplace, and Whittaker transforms. Some interesting special cases of the main finding are also addressed and demonstrated to be related to some well-known ones.

Recently, Liu and Wang generalized Appell's and Lauricella's hypergeometric functions. Motivated by the work of Liu and Wang, the main object of this paper is to present new generalizations of Appell's and Lauricella's hypergeometric functions. Some integral representations, transformation formulae, differential formulae and recurrence relations are obtained for these new generalized Appell's and Lauricella's functions.

Recently, Srivastava et al. [18] introduced the incomplete Pochhammer symbol and studied some fundamental properties and characteristics of a family of potentially useful incomplete hypergeometric functions. Here we introduce the incomplete Lauricella function <TEX>${\gamma}_D^{(n)}$</TEX> and <TEX>${\Gamma}_D^{(n)}$</TEX> of n variables, and investigate certain properties of the incomplete Lauricella functions, for example, their various integral representations, differential formula and recurrence relation, in a rather systematic manner. Some interesting special cases of our main results are also considered.

We present a novel generalized Mittag-Leffler-type function of arbitrary order in this study. We look into its fundamental characteristics, such as differential formulas, recurrence relations, integral representations, the Euler, Laplace, Mellin, Whittaker, and Mellin-Barnes transforms. Additionally, we express it using the H-function, generalized hypergeometric function, and Fox-Wright function. Further, for this generalized Mittag-Leffler-type function of arbitrary order, we create fractional integral and differential operators. Also, we derive several intriguing special instances of our fundamental results.

In the recent years, various authors introduced different generalizations of the Hurwitz–Lerch zeta function and discussed its various properties. The main aim of our study is to introduce a new bicomplex generalization of the Hurwitz–Lerch zeta function using the new generalized form of the beta function that involves the Appell series and Lauricella functions. The new bicomplex generalization of the Hurwitz–Lerch zeta function reduces to some already known functions like the Hurwitz–Lerch zeta function, Hurwitz zeta function, Riemann zeta function and polylogarithmic function. Its different properties such as recurrence relation, summation formula, differentiation formula, generating relations and integral representations are investigated. All results induced are general in nature and reducible to already known results. As an application of the new bicomplex generalization of the Hurwitz–Lerch zeta function, we have developed a new generalized form of fractional kinetic equation and obtained its solution using the natural transform.

The Mittag-Leffler~\cite{mit} function ascends naturally in the solution of fractional order differential and integral equations, and exclusively in the studies of fractional generalizing of kinetic equation, random walks, L\'{e}vy flights, super-diffusive transport and in the study of complex systems. In the present investigation, the authors define a new class of meromorphic functions defined in the punctured unit disk $\Delta^*:= \{z\in\mathbb{C}: 0&lt;|z|&lt;1\}$ based on Mittag-Leffler function denoted by $\mathfrak{M}^{\tau,\kappa}_{\varsigma,\varrho}(\vartheta,\wp)$. We discuss its characteristic properties like coefficient inequalities, growth and distortion inequalities, as well as closure results for $f\in\mathfrak{M}^{\tau,\kappa}_{\varsigma,\varrho}(\vartheta,\wp)$extensively. Properties of a certain integral operator and its inverse defined on the new class $\mathfrak{M}^{\tau,\kappa}_{\varsigma,\varrho}(\vartheta,\wp)$ are also discussed. Coefficient inequalities, growth and distortion inequalities, as well as closure results are obtained. We also prove a Property using an integral operator and its inverse defined on the new class.We also establish some results concerning neighborhoods and the partial sums of meromorphic functions in this new class. We also state some new subclasses and its characteristic properties by specializing the parameters which are new and not studied in association with Mittag-Leffler functions.