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For T n (x) = cos n arccos x , x ∈ [ - 1 , 1 ] , the n-th Chebyshev polynomial of the first kind, we study the quantity τ n , k : = | T n (k) (ω n , k) | T n (k) (1) , 1 ≤ k ≤ n - 2 , where T n (k) is the k-th derivative of T n and ω n , k is the largest zero of T n (k + 1) . Since the absolute values of the local extrema of T n (k) increase monotonically towards the end-points of [ - 1 , 1 ] , the value τ n , k shows how small is the largest critical value of T n (k) relative to its global maximum T n (k) (1) . This is a continuation of our (joint with Alexei Shadrin) paper "On the largest critical value of T n (k) ", SIAM J. Math. Anal.50(3), 2018, 2389–2408, where upper bounds and asymptotic formuae for τ n , k have been obtained on the basis of the Schaeffer–Duffin pointwise upper bound for polynomials with absolute value not exceeding 1 in [ - 1 , 1 ] . We exploit a 1996 result of Knut Petras about the weights of the Gaussian quadrature formulae associated with the ultraspherical weight function w λ (x) = (1 - x 2) λ - 1 / 2 to find an explicit (modulo ω n , k ) formula for τ n , k 2 . This enables us to prove a lower bound and to refine the previously obtained upper bounds for τ n , k . The explicit formula admits also a new derivation of the asymptotic formula approximating τ n , k for n → ∞ . The new approach is simpler, without using deep results about the ordinates of the Bessel function, and allows to better analyze the sharpness of the estimates. [ABSTRACT FROM AUTHOR]

In this paper, taking into account the intriguing recent results of Rabotnov functions, Poisson functions, Bessel functions and Wright functions, we consider a new comprehensive subclass O μ (Δ 1 , Δ 2 , Δ 3 , Δ 4) of univalent functions defined in the unit disk Λ = { τ ∈ C : τ < 1 } . More specifically, we investigate some sufficient conditions for Rabotnov functions, Poisson functions, Bessel functions and Wright functions to be in this subclass. Some corollaries of our main results are given. The novelty and the advantage of this research could inspire researchers of further studies to find new sufficient conditions to be in the subclass O μ (Δ 1 , Δ 2 , Δ 3 , Δ 4) not only for the aforementioned special functions but for different types of special functions, especially for hypergeometric functions, Dini functions, Sturve functions and others. [ABSTRACT FROM AUTHOR]

Geometric function theory has extensively explored the geometric characteristics of analytic functions within symmetric domains. This study analyzes the geometric properties of a specific class of analytic functions employing confluent hypergeometric functions and generalized Bessel functions of the first kind. Specific constraints are imposed on the parameters to ensure the inclusion of the confluent hypergeometric function within the analytic function class. The coefficient bound of the class is used to determine the inclusion properties of integral operators involving generalized Bessel functions of the first kind. Different results are observed for these operators, depending on the specific values of the parameters. The results presented here include some previously published findings as special cases. [ABSTRACT FROM AUTHOR]

A one-dimensional model for fluid and solute transport in poroelastic materials (PEMs) is studied. Although the model was recently derived and some exact solutions, in particular steady-state solutions and their applications, were studied, special cases occurring when some parameters vanish were not analysed earlier. Since the governing equations are nonintegrable in nonstationary cases, the Lie symmetry method and modern tools for solving ODE systems are applied in order to construct time-dependent exact solutions. Depending on parameters arising in the governing equations, several special cases with new Lie symmetries are identified. Some of them have a highly nontrivial structure that cannot be predicted from a physical point of view or using Lie symmetries of other real-world models. Applying the symmetries obtained, multiparameter families of exact solutions are constructed, including those in terms of elementary and special functions (hypergeometric, Whittaker, Bessel and modified Bessel functions). A possible application of the solutions obtained is demonstrated, and it is shown that some exact solutions can describe (at least qualitatively) the solute transport in PEM. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate analytical and numerical methods for solving relevant boundary value problems. [ABSTRACT FROM AUTHOR]

We sum in a closed form the Sneddon-Bessel series ∑∞ m=1 Jα(xjm,v)Jβ(yjm,v) /j m,v2n+α+β-2v+2 Jv+1(jm,v)², where 0 < x, 0 < y, x + y < 2, n is an integer, α, β, v ε C\{-1,-2, ... } with 2 Re v < 2n + 1 + Re α + Re β and {jm,v}m≥0 are the zeros of the Bessel function Jv of order v. In most cases, the explicit expressions for these sums involve hypergeometric functions pFq. As an application, we prove some extensions of the Kneser-Sommerfeld expansion. For instance, we show that ∑∞ m=1 j v-β m,v Jv (xjm,v)Jβ (yjm,v) (j²m,v-z²)Jv+1(jm,v)² = πJβ (yz)/4zβ-v Jv (z) (Yv (z)Jv (xz) - Jv (z)Yv (xz)), if Re v < Re β + 1 and 0 < y ≤ x, x + y < 2 (here, Yv denotes the Bessel function of the second kind), which becomes the Kneser-Sommerfeld expansion when β = v. [ABSTRACT FROM AUTHOR]

The Bessel function of the first kind J N k x is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind I N k x . The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for J N k x useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving   1 F 2 hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes 1 / 2 i 3 j 5 k 7 l 11 m 13 n 17 o 19 p multiplying powers of the coefficient k. [ABSTRACT FROM AUTHOR]

In this article, we get certain integral representations of the multi-index Wright generalized Bessel function by making use of the extended beta function. This function is presented as a part of the generalized Bessel–Maitland function obtained by taking the extended fractional derivative of the generalized Bessel–Maitland function developed by Özarsalan and Özergin [M. Ali Özarslan and E. Özergin, Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Math. Comput. Model. 52 2010, 9–10, 1825–1833]. In addition, we demonstrate the exciting connections of the multi-index Wright generalized Bessel function with Laguerre polynomials and Whittaker function. Further, we use the generalized Wright hypergeometric function to calculate the Mellin transform and the inverse of the Mellin transform. [ABSTRACT FROM AUTHOR]

In this work, the geometric nature of solutions to two second-order differential equations, z y ′ ′ (z) + a (z) y ′ (z) + b (z) y (z) = 0 and z 2 y ′ ′ (z) + a (z) y ′ (z) + b (z) y (z) = d (z) , is studied. Here, a (z) , b (z) , and d (z) are analytic functions defined on the unit disc. Using differential subordination, we established that the normalized solution F (z) (with F(0) = 1) of above differential equations maps the unit disc to the domain bounded by the leminscate curve 1 + z . We construct several examples by the judicious choice of a (z) , b (z) , and d (z) . The examples include Bessel functions, Struve functions, the Bessel–Sturve kernel, confluent hypergeometric functions, and many other special functions. We also established a connection with the nephroid domain. Directly using subordination, we construct functions that are subordinated by a nephroid function. Two open problems are also suggested in the conclusion. [ABSTRACT FROM AUTHOR]

The aim of this work is to establish a connection between Bohr's radius and the analytic and normalized solutions of two differential second-order differential equations, namely y ″ (z) + a (z) y ′ (z) + b (z) y (z) = 0 and z 2 y ″ (z) + a (z) y ′ (z) + b (z) y (z) = d (z) . Using differential subordination, we find the upper bound of the Bohr and Rogosinski radii of the normalized solution F (z) of the above differential equations. We construct several examples by judicious choice of a (z) , b (z) and d (z) . The examples include several special functions like Airy functions, classical and generalized Bessel functions, error functions, confluent hypergeometric functions and associate Laguerre polynomials. [ABSTRACT FROM AUTHOR]

Sufficient conditions are determined on the parameters such that the generalized and normalized Bessel function of the first kind, which is an elementary transform of the hypergeometric function and other related functions belong to subclasses of starlike and convex functions defined in the unit disk associated with the exponential mapping. Several differential subordination implications are derived for analytic functions involving Bessel function and the operator introduced by Baricz et al. [Differential subordinations involving generalized Bessel functions, Bull. Malays. Math. Sci. Soc. 38(3) (2015), 1255–1280]. These results are obtained by constructing suitable class of admissible functions. Examples involving trigonometric and hyperbolic functions are provided to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

This paper studies a subclass of the class of generalized hyperbolic distribution called the semi-hyperbolic distribution. We obtain analytical expressions for the cumulative distribution function and, specifically, their first and second lower partial moments. Using the received formulas, we compute the value at risk, the expected shortfall, and the semivariance in the semi-hyperbolic model of the financial market. The formulas depend on the values of generalized hypergeometric functions and modified Bessel functions of the second kind. The research illustrates the possibility of analysis of generalized hyperbolic models using the same methodology as is employed for the well-established variance-gamma model. [ABSTRACT FROM AUTHOR]

Some well-known results of Grönwall on logarithmic derivative of modified Bessel functions of the first kind concerning exponential bounds are extended to Whittaker functions of the first and second kind M_{\kappa,\mu } and W_{\kappa,\mu }. Moreover, a complete monotonicity result is proved for the logarithmic derivative of the Whittaker function W_{\kappa,\mu }, and some monotonicity results with respect to the parameters and argument are shown for the logarithmic derivative of M_{\kappa,\mu }. The results extend and complement the known results in the literature about modified Bessel functions of the first and second kind. [ABSTRACT FROM AUTHOR]

An attempt is made to define Modified Generalized Bessel Function, and Modified Generalized Bessel Matrix Function. Some properties have also been discussed. [ABSTRACT FROM AUTHOR]

Two integral transforms involving Gaussian hypergeometric functions in the Kernel, which are generalization of classical Riemann-Liouville and Erdeyi-Kober fractional integral are considered and Applied on the Bessel functions of the first kind and thus two main results are derived, which are given in the form of two theorems. These two main results are obtained in terms of generalized wright functions, which are also expressed in terms of generalized hypergeometric functions, Corollaries & special cases are also considered. [ABSTRACT FROM AUTHOR]

In this article, we obtain an explicit general solution of the Schrödinger equation with the squared secant potential ν (ν + 1) sec 2 x in terms of elementary functions for non-negative integer value of ν = n . Alternatively, we provide a general solution in terms of the Gauss hypergeometric function for any parameter value ν . Then, we derive some hypergeometric identities by comparing these two sets of solutions when ν is a non-negative integer. With the help of these hypergeometric formulas, we derive several existing explicit representations as well as new ones for some special functions and orthogonal polynomials including the Legendre, Ferrers, and the Bessel functions, the Jacobi and the Lommel polynomials containing specific parameters. [ABSTRACT FROM AUTHOR]

Spherical Bessel functions (sBFs) appear commonly in many areas of physics wherein there is both translation and rotation invariance, and often integrals over products of several arise. Thus, analytic evaluation of such integrals with different weighting functions (which appear as toy models of a given physical observable, such as the galaxy power spectrum) is useful. Here, we present a generalization of a recursion-based method for evaluating such integrals. It gives relatively simple closed-form results in terms of Legendre functions (for the exponentially damped case) and Gamma, incomplete Gamma, and hypergeometric functions (for the Gaussian-damped case). We also present a new, non-recursive method to evaluate integrals of products of sBFs with Gaussian damping in terms of incomplete Gamma functions and hypergeometric functions. [ABSTRACT FROM AUTHOR]

We consider operator hypergeometric functions 1F2(∙) and 2F3(∙) constructed by an unbounded operator. Using these functions, we solve Cauchy problems for singular integro-differential equations. A new pair of similar operators is given. [ABSTRACT FROM AUTHOR]

The present paper is devoted to derive a generalized Oberhettinger type integral formula. The derived form of an integral involving the product of the four-parameter Bessel function and Jacobi polynomial. The outcomes are expressed in terms of the Kampé de Fériet and Srivastava and Daoust functions. Also, four Corollaries of the both Theorems are derived in terms of the Kampé de Fériet and Srivastava and Daoust functions. Some of the significant particular cases are also determined. Furthermore, we drive an interesting relationship between Kampé de Fériet and Srivastava and Daoust functions. [ABSTRACT FROM AUTHOR]

This paper deals with the evaluation of certain integral transforms involving the product of certain Appell and Bessel functions with a weight e - γ ⁢ t 2 . The transformations of these integrals are evaluated in terms of the Appell, Kampé de Fériet and the triple hypergeometric functions. As an application, we studied propagation of generalized Humbert–Gaussian beams (GHGBs) and hypergeometric-Gaussian beams (HyGGBs) in turbulent atmosphere and through an ABCD paraxial optical system. The evaluation of these integral transforms has initiated a great interest in mathematical physics and its applications to laser physics and linear or non-linear optics. [ABSTRACT FROM AUTHOR]

We calculate some infinite sums containing the digamma function in closed form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as parameter differentiation formulas for the beta incomplete function, reduction formulas of 3 F 2 hypergeometric functions, or a definite integral which does not seem to be tabulated in the most common literature. As an application of certain sums involving the digamma function, we calculated some reduction formulas for the parameter differentiation of the Mittag–Leffler function and the Wright function. [ABSTRACT FROM AUTHOR]

This paper analyses a single buffer fluid queueing system where the fluid flow into the buffer is regulated by a queue with impatient customers and exponential vacations. The governing differential equations of the fluid queueing system are solved using generating function method and continued fraction methodology. Analytical expressions for the stationary distribution of the buffer content are obtained in terms of Bessel functions and confluent hypergeometric functions by employing Laplace transforms. The average buffer content of the system under consideration is also obtained. [ABSTRACT FROM AUTHOR]

Named essentially after their close relationship with the modified Bessel function K ν (z) of the second kind, which is known also as the Macdonald function (or, with a slightly different definition, the Basset function), the so-called Bessel polynomials y n (x) and the generalized Bessel polynomials y n (x ; α , β) stemmed naturally in some systematic investigations of the classical wave equation in spherical polar coordinates. Our main purpose in this invited survey-cum-expository review article is to present an introductory overview of the Bessel polynomials y n (x) and the generalized Bessel polynomials y n (x ; α , β) involving the asymmetric parameters α and β. Each of these polynomial systems, as well as their reversed forms θ n (x) and θ n (x ; α , β) , has been widely and extensively investigated and applied in the existing literature on the subject. We also briefly consider some recent developments based upon the basic (or quantum or q-) extensions of the Bessel polynomials. Several general families of hypergeometric polynomials, which are actually the truncated or terminating forms of the series representing the generalized hypergeometric function r F s with r symmetric numerator parameters and s symmetric denominator parameters, are also investigated, together with the corresponding basic (or quantum or q-) hypergeometric functions and the basic (or quantum or q-) hypergeometric polynomials associated with r Φ s which also involves r symmetric numerator parameters and s symmetric denominator parameters. [ABSTRACT FROM AUTHOR]

Integral form expressions are obtained for the Mathieu-type series and for their associated alternating versions, the terms of which contain a (p , ν) -extended Gauss hypergeometric function. Contiguous recurrence relations are found for the Mathieu-type series with respect to two parameters, and finally, particular cases and related bounding inequalities are established. [ABSTRACT FROM AUTHOR]

Various relations including differentiation formulas, connection with hypergeometric functions, integral representations, formulas of summation and series representations for three types of Neumann polynomials are derived. [ABSTRACT FROM AUTHOR]

Bicomplex convolutional neural networks (BCCNN) are a natural extension of the quaternion convolutional neural networks for the bicomplex case. As it happens with the quaternionic case, BCCNN has the capability of learning and modelling external dependencies that exist between neighbour features of an input vector and internal latent dependencies within the feature. This property arises from the fact that, under certain circumstances, it is possible to deal with the bicomplex number in a component-wise way. In this paper, we present a BCCNN, and we apply it to a classification task involving the colourized version of the well-known dataset MNIST. Besides the novelty of considering bicomplex numbers, our CNN considers an activation function a Bessel-type function. As we see, our results present better results compared with the one where the classical ReLU activation function is considered. [ABSTRACT FROM AUTHOR]

We use the tridiagonal representation approach to solve the radial Schrödinger equation for the continuum scattering states of the Coulomb problem in a complete basis set of discrete Bessel functions. Consequently, we obtain a new representation of the confluent hypergeometric function as an infinite sum of Bessel functions, which is numerically very stable and more rapidly convergent than another well-known formula. [ABSTRACT FROM AUTHOR]

We compute the Laplace transforms of some integral functionals of the three-dimensional Bessel process in terms of modified Bessel functions, Gauss' hypergeometric functions, and confluent hypergeometric functions. Some new results are obtained, and several established results, such as Dufresne's perpetuity and a particular case of its translated version, are recovered. In particular, we derive the Laplace transform of the two-dimensional random variable (∫ 0 ∞ (exp (ρ t) − η) − 1 d t , ∫ 0 ∞ (exp (ρ t) − η) − 2 d t) , where ρ is a three-dimensional Bessel process and η ≤ 1. A crucial step of the derivation is to construct a martingale, as performed in the Bass solution of the Skorokhod embedding problem, with respect to an enlarged filtration. [ABSTRACT FROM AUTHOR]

An attempt is made to obtain some properties of Extended Generalized Bessel function. Extended Modified Bessel Function has also been discussed. [ABSTRACT FROM AUTHOR]

Our purpose in the present investigation is to study certain geometric properties such as the close-to-convexity, convexity, and starlikeness of the hypergeometric function z 1 F 2 (a ; b , c ; z) in the open unit disk. The usefulness of the main results for some familiar special functions like the modified Sturve function, the modified Lommel function, the modified Bessel function, and the F 1 0 (− ; c ; z) function are also mentioned. We further consider a boundedness property of the function F 2 1 (a ; b , c ; z) in the Hardy space of analytic functions. Several corollaries and special cases of the main results are also pointed out. [ABSTRACT FROM AUTHOR]

Orthogonal polynomials for the weight x ν exp ⁡ (− x − t / x) , x , t > 0 , ν ∈ R are investigated without the use of the Chen-Ismail ladder operators approach. In this part we derive explicit representations, recurrence relations for coefficients, the generating function and Rodrigues-type formula. [ABSTRACT FROM AUTHOR]

Introduction/purpose: Certain integrals involving the generalized MittagLeffler function with different types of polynomials are established. Methods: The properties of the generalized Mittag-Leffler function are used in conjunction with different kinds of polynomials such as Jacobi, Legendre, and Hermite in order to evaluate their integrals. Results: Some integral formulae involving the Legendre function, the Bessel Maitland function and the generalized hypergeometric functions are derived. Conclusions: The results obtained here are general in nature and could be useful to establish further integral formulae involving other kinds of polynomials. [ABSTRACT FROM AUTHOR]

Rapidly convergent series are derived to efficiently evaluate a class of integrals involving the product of spherical Bessel functions of the first kind occurring in acoustic and electromagnetic scattering from circular disks and apertures. Depending on the involved parameters, the series can be further reduced to closed-form expressions in terms of generalized hypergeometric functions or Meijer G-functions, which can immediately be evaluated through current mathematical toolboxes. Numerical results are provided showing that the accuracy of the series representation can easily be controlled and that the proposed solutions are at least 1000 times faster than specific quadrature schemes which, in general, have to deal with irregularly oscillating and slowly decaying functions. • Efficient evaluation of improper integrals involving the product of spherical Bessel functions. • Efficient expressions in terms of rapidly convergent series and/or in terms of special functions. • Speed-up factor larger than 1000 with respect to the most updated quadrature routines. • Detailed numerical comparisons shows the accuracy and the efficiency. • Possibility to extend the approach to other integrals occurring in different field of physics. [ABSTRACT FROM AUTHOR]

In this article, we derived conditions on the coefficient functions a (z) and b (z) of the differential equations y ″ (z) + a (z) y ′ (z) + b (z) y (z) = 0 and z 2 y ″ (z) + a (z) z y ′ (z) + b (z) y (z) = 0 , such their solution f (z) with normalization f (0) = 0 = f ′ (0) − 1 is starlike in the lemniscate domain, equivalently z f ′ (z) / f (z) ≺ 1 + z . We provide several examples with graphical presentations for a clear view of the obtained results. [ABSTRACT FROM AUTHOR]

Orthogonal polynomials for the weight x ν exp ⁡ (− x − t / x) , x , t > 0 , ν ∈ R are investigated without the use of the Chen–Ismail ladder operators approach. Differential and differential–difference equations are deduced. [ABSTRACT FROM AUTHOR]

A method related to Wronskians has recently been applied to derive many indefinie integrals involving Bessel functions and associated Legendre functions. Here the same method is applied to derive indefinite integrals of Whittaker functions and Gauss hypergeometric functions. All the integrals presented here appear to be new and are derived either directly from Wronskians or from a differential equation related to Wronskians. All the results presented have been checked numerically using Mathematica. [ABSTRACT FROM AUTHOR]

In this work, we define an extension of the k-Wright ((k;τ)-Gauss) hypergeometric matrix function and obtain certain properties of this function. Further, we present this function to achieve the solution of the fractional kinetic equations. [ABSTRACT FROM AUTHOR]

Special functions can be defined in different ways such as Rodrigue's formulae, generating functions, summation formulae, integral representations et cetera, but it is usually shown to be expressible as a series, because this is frequently the most practical way to obtain numerical values for the functions. In this paper, the explicit representations of certain mixed special functions related to the Bessel functions are obtained. A Laurent type hypergeometric generating relation is derived using series rearrangement technique. Some special cases are obtained as generating function of the known mixed type relatives of the Bessel functions. Finally explicit forms of these mixed type special functions are obtained as applications. [ABSTRACT FROM AUTHOR]

A new representation for the probability density function (PDF) of a single real sinusoid in additive white Gaussian noise is presented. It has the form of an infinite series of the exponentially scaled modified Bessel functions of the first kind of integer order. The new PDF is an alternative to the classical PDF's expressed in terms of the derivatives of the error function, the confluent hypergeometric function of the first kind, and the exponentially scaled Hermite polynomials. [ABSTRACT FROM AUTHOR]

Many indefinite integrals are derived for Bessel functions and associated Legendre functions from particular transformations of their differential equations which are closely linked to Wronskians. A large portion of the results for Bessel functions is known, but all the results for associated Legendre functions appear to be new. The method can be applied to many other special functions. All results have been checked by differentiation using Mathematica. [ABSTRACT FROM AUTHOR]

Introduction/purpose: The aim of this article is to establish integral transforms of the generalized Lommel-Wright function. Methods: These transforms are expressed in terms of the Wright Hypergeometric function. Results: Integrals involving the trigonometric, generalized Bessel function and the Struve functions are obtained. Conclusions: Various interesting transforms as the consequence of this method are obtained. [ABSTRACT FROM AUTHOR]

Recently, Dixit et al. [24] established a very elegant generalization of Hardy's theorem concerning the infinitude of zeros that the Riemann zeta function possesses at its critical line. By introducing a general transformation formula for the theta function involving the Bessel and modified Bessel functions of the first kind, we extend their result to a class of Dirichlet series satisfying Hecke's functional equation. In the process, we also find new generalizations of classical identities in Analytic number theory. [ABSTRACT FROM AUTHOR]

An efficient and accurate method for calculating the sound radiated by a baffled circular rigid piston is using spherical harmonics, and the solution is a series containing the integral of spherical Bessel functions. The integral is usually calculated with the generalized hypergeometric functions in existing literatures, which shows poor convergence at middle and high frequencies due to the overflow and the loss of significant figures. A rigorous and closed form solution of the integral is derived in this paper based on the recurrence method, which is accurate in the whole frequency range and thousands of times faster than the existing methods. It is shown that the proposed method can be extended for the calculation of the sound radiated by a baffled piston and an unbaffled resilient disk with axisymmetric velocity and pressure profiles, respectively, and some baffled rotating sources where the velocity profile is asymmetric. [ABSTRACT FROM AUTHOR]

Discrete analogues of the classical Fourier-Jacobi transform are introduced and investigated. It involves series and integrals with respect to parameters of the Gauss hypergeometric function 2 F 1 (a + i n / 2 , a − i n / 2 ; c ; − x 2) , x > 0 , n ∈ N , a , c > 0 , i is the imaginary unit. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals are established. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove Laplace transform of some special functions in term of generalized Meijer G-functions. Some properties of generalized Meijer G-functions will be discussed. We investigate the Laplace transform of different hypergeometric functions in the form of generalized Meijer G-functions and hypergeometric functions. We derive Laplace transform of Bessel k-functions, hyper-Bessel k-functions, incomplete gamma k-function, sine k-integral, sine hyperbolic k-integral, Kelvin k-function in the form of generalized Meijer G-functions. In fact, we provide new approach to find Laplace transform of said functions. [ABSTRACT FROM AUTHOR]

The present work deals with the mathematical investigation of some generalizations of Bessel functions. The main motive of this paper is to show that the generating function can be employed efficiently to obtain certain results for special functions. The complex form of Bessel functions is introduced by means of the generating function. Certain enthralling properties for complex Bessel functions are investigated using the generating function method. By considering separately the real and the imaginary part of complex Bessel functions, we get respectively cosine-Bessel functions and sine-Bessel functions for which several novel identities and Jacobi–Anger expansions are established. Also, the generating function of degenerate Bessel functions is investigated and certain novel identities are obtained for them. A hybrid form of degenerate Bessel functions, namely, of degenerate Fubini–Bessel functions, is constructed using the replacement technique. Finally, the explicit forms of the real and the imaginary part of complex Bessel functions are established by a hypergeometric approach. [ABSTRACT FROM AUTHOR]

For \mu,\nu >-1, we consider the Bessel series \[ U_{\mu,\nu }^{\mathfrak {a}}(x) = \frac {2^\mu \Gamma (\mu +1)}{x^\mu } \sum _{m\ge 1} \frac {a_m}{j_{m,\nu }^{\mu +1/2}} J_\mu (j_{m,\nu } x), \] where (j_{m,\nu })_{m\ge 1} are the positive zeros of J_\nu and \mathfrak {a} = (a_m)_{m\ge 1} is a sequence of real numbers satisfying \sum _{m\ge 1} {|a_m|}/{j_{m,\nu }^{\mu +1/2}} < +\infty. We propose a method for computing in a closed form the sum of the Bessel series U_{\mu,\nu }^{\mathfrak {a}} assuming that for a particular value \eta of the parameter \mu a closed expression for U_{\eta,\nu }^{\mathfrak {a}} as a power series of x (not necessarily with integer exponents) is known. We illustrate the method with some examples. One of them is related to the sine coefficients of the function 1-x^s, s>-1. The closed form of the sum is then given in terms of a generalization of the Bernoulli numbers. [ABSTRACT FROM AUTHOR]

In this paper, we use an identity connecting a modified q -Bessel function and a 1 ϕ 1 function to give m -versions of entries in the Lost Notebook of Ramanujan. We also establish an identity which gives an m -version of a partition identity. We prove new relations and identities involving theta functions, the Ramanujan function, the Stieltjes–Wigert, q -Lommel and q -Bessel polynomials. We introduce and study q -analogues of the spherical Bessel functions. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the incomplete extended Gamma and Beta functions with matrix parameters; then, we establish some different properties for these new extensions. Furthermore, we give a specific application for the incomplete Bessel matrix function by using incomplete extended Gamma and Beta functions; at last, we construct the relation between the incomplete confluent hypergeometric matrix functions and incomplete Bessel matrix function. [ABSTRACT FROM AUTHOR]