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The propagation of a nonlinearly elastic longitudinal plane displacement wave for a symmetric initial wave profile described by a Gaussian function and an asymmetric initial profile described by a Whittaker function is analyzed. The basic novelty is that the evolution of waves is analyzed by the approximate methods by taking into account the first three approximations. The harmonic wave is analyzed only to compare with the new results for the bell-shaped wave. Some significant differences between the evolutions of waves are shown. Common to these profiles is the distortion of the initial profile due to the nonlinear self-interaction of the wave. The bell-shaped (symmetrical profile) wave retains symmetry when moving in a nonlinearly elastic body. For some initial sets of parameters, this wave does not change the wavelength initially and only shows a tendency towards the formation of two humps instead of one when we take into account the second conditional harmonic (two bell-shaped waves form and adjoin each other and halve wavelength) and the sinking of the left and elevation of the right hump when we take into account the third harmonic. The asymmetric profile described by Whittaker function retains the wavelength and asymmetry when the second approximation of the second harmonic is taken into account, but the amplitude increases rapidly. With additional allowance for the third harmonic, two asymmetric humps are formed, the humps resemble the evolution of the symmetric profile when the second harmonic is taken into account. Common to the evolutions of the symmetric and asymmetric profiles is the distortion of the initial wave profile: the formation of two symmetric humps when the second harmonic is taken into account in the analysis of the Gaussian function and two asymmetric humps when the third harmonic is taken into account in the analysis of the Whittaker function. Taking the third harmonic into account for a symmetric profile makes the two humps asymmetrical for symmetric and asymmetric profiles, only in these two profiles, the asymmetry is different.

In the present problem, an attempt has been made to study the propagation of Rayleigh waves in an incompressible medium with polynomial variation (m) of rigidity over an incompressible half-space under rigid layer. Instead of using the Whittaker function, the expansion formula proposed by Newlands has been used for a better result in shallow depth. The velocity equation has been calculated and the results are shown in figures. The study in the assumed medium, the authors obtained that the phase velocity of Rayleigh waves increases except for the polynomial variation of rigidity m = 1, 2 and 3.

In this paper, we propose a new scheme for the temporal discretization of the new Caputo–Hadamard fractional derivative in the form of incomplete gmma function via Whittaker M function. We derive the truncation error for the new scheme. Hence, this discretization was used to solve numerically fractional ordinary differential equation and fractional partial differential equation in the Caputo–Hadamard sense for the order, $$\alpha \in (0,1)$$ . The numerical results show that the method is highly effective and efficient.

In moderate to strong oceanic turbulence, a differential phase-shift-keying (DPSK) modulated underwater wireless optical communication (UWOC) system with an aperture receiver is presented. The analytical expressions for mean bit error rate (BER) of UWOC system over moderate to strong oceanic turbulence are derived based on Gamma–Gamma channel model by the help of the Whittaker function. The variations of mean BER are investigated versus modulation methods, beam types, aperture diameters as well as four oceanic turbulence parameters, i.e., the rate of dissipation of kinetic energy per unit mass of fluid, the rate of dissipation of mean-squared temperature, the ratio of temperature to salinity contributions to the refractive index spectrum and the kinetic viscosity.

We deduce a product formula for the Whittaker $W$ function whose kernel does not depend on the second parameter. Making use of this formula, we define the positivity-preserving convolution operator associated with the index Whittaker transform, which is seen to be a direct generalization of the Kontorovich-Lebedev convolution. The mapping properties of this convolution operator are investigated; in particular, a Banach algebra property is established and then applied to yield an analogue of the Wiener-L\'evy theorem for the index Whittaker transform. We show how our results can be used to prove the existence of a unique solution for a class of convolution-type integral equations., Comment: 24 pages. Accepted for publication in Journal of Mathematical Analysis and Applications

The problem of constructing asymptotics describing far-field internal gravity waves generated by an oscillating point source of perturbations moving in a vertically semi-infinite stratified layer of variable buoyancy is considered. For a model distribution of the buoyancy frequency, analytical solutions of the main boundary value problem are obtained, which are expressed in terms of Whittaker functions. An integral representation for the Green’s function is obtained, and asymptotic solutions are constructed that describe the amplitude-phase characteristics of internal gravity wave fields in a semi-infinite stratified medium with a variable buoyancy frequency far away from the perturbation source.

We propose Whittaker function approach as a theoretical method for finding exact solutions of a quantum mechanical system placed in the Kratzer potential. We then show that the effect of an external uniform magnetic field on this system can be satisfactorily determined using variational method. By following the one-step treatment suggested in this study, we increase the reliability and the accuracy of the solutions of Schrodinger equation for a quantum mechanical system placed in potential energy and perturbed by a uniform magnetic field that proves to be useful in modelling physical phenomena. We find that the achieved numerical and analytical results agree very well with those already published and those calculated using the Numerov method.

In this paper, we aim to establish a closed form for an integral transform involving the product of two Bessel functions of the first kind and the Whittaker function. This integral transform is evaluated in terms of the Lauricella’s triple hypergeometric function, which reduces to the generalized Kampe de Feriet function and is interesting to generate some new laser beams and to study the propagation of these waves through free space and atmospheric and maritime turbulent. Besides, some particular cases are derived by using the relations between the Whittaker function and other special functions and orthogonal polynomials. A novel beams family, called “ Lauricella beams ” is introduced and the propagation of Laguerre–Bessel–Gaussian beam through an ABCD optical system is studied.

The present work concerns the general stream function solution of the Brinkman equation in parabolic cylindrical coordinates, arising in the study of fluid flow through porous medium. Analytical stream function solutions of this equation are available in the coordinates (Cartesian, cylindrical polar, spherical polar and prolate spheroidal coordinates). Stream function solution of the Stokes equation in parabolic cylindrical coordinates is also investigated analytically. The parabolic cylinder functions are a class of functions which are the solution of Weber differential equation. A transformation of parabolic cylinder function into the Whittaker function is used. Method of inverse operator is applied to obtain particular integral in solving the Stokes equation. Explicit expressions of velocity components and vorticity are also reported.

Discrete analogs of the index Whittaker transform are introduced and investigated. It involves series and integrals with respect to a second parameter of the Whittaker function $$W_{\mu , {i n} }(x), \ x >0, \ \mu \in \mathbb {R}, \ n \in \mathbb {N}, \ i $$ is the imaginary unit. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals are established.

In the present paper, investigation has been made to study the propagation character of Love wave in an orthotropic substratum being layered doubly over anisotropic porous mantle. Separate expressions for displacement components have been derived for each of the three layer and suitable boundary conditions have been imposed to derive the dispersion equation of the wave in a closed form in terms of Whittaker function. It has been found that inhomogeneity parameter associated with the rigidity and the density of the layered medium has a special bearing on the Love wave propagation. Orthotropic properties and initial stresses of the media also affects the velocity of the wave to a great extent has been shown graphically.

We introduce vector solitary waves in two-component Bose–Einstein condensates with spatially modulated nonlinearity coefficients and a harmonic trapping potential. Using the self-similarity method, novel vector solitary waves are built with the help of Whittaker function, including multipole solutions and necklace rings. The stability of vortex soliton pairs is examined by direct numerical simulation; the results show that a new class of stable low-order vortex soliton pairs with n = 2 and m ≤ 3 can be supported by the spatially modulated interaction in the harmonic trap. Higher order vector-vortex soliton is found unstable over prolonged distances.

In this article, we introduce a further generalizations of the confluent hypergeometric function and Whittaker function by introducing an extra parameter in the extended con uent hypergeometric function dened by Parmar [15]. We also investigate some integral representations, some integral transforms, differential formulas and recurrence relations of these new generalizations

The non-isothermal analysis of reactions with a constant heating rate involves a temperature integral. This integral is popularized as the Arrhenius integral, but when the pre-exponential factor of the Arrhenius equation depends on temperature with a power-law relationship, the integral is known as the general temperature integral. In this paper, it is shown that when the activation energy and exponent of power-law assumed to be constant, these integrals can be expressed and evaluated by several kinds of special functions and using commercial or free softwares. The special functions include exponential integral function, incomplete gamma function, confluent hypergeometric function, Whittaker function and generalized hypergeometric function. New rational approximations can be obtained by expansion of the special functions. MATLAB and GNU Octave were used to compute the special function models as well as numerical integration. The accuracy of these methods was examined by variable precision arithmetic. For both softwares, the special function models showed higher accuracies compared to the numerical integration methods and the approximation functions. In addition, new iterative isoconversional method for constant activation energy and exponent of power-law has been proposed that shows very high accuracy in calculation of the activation energy.

We derive the time-dependent probability distribution for the number of infected individuals at a given time in a stochastic Susceptible-Infected-Susceptible (SIS) epidemic model. The mean, variance, skewness, and kurtosis of the distribution are obtained as a function of time. We study the effect of noise intensity on the distribution and later derive and analyze the effect of changes in the transmission and recovery rates of the disease. Our analysis reveals that the time-dependent probability density function exists if the basic reproduction number is greater than one. It converges to the Dirac delta function on the long run (entirely concentrated on zero) as the basic reproduction number tends to one from above. The result is applied using published COVID-19 parameters and also applied to analyze the probability distribution of the aggregate number of COVID-19 cases in the United States for the period: January 22, 2020-March 23, 2021. Findings show that the distribution shifts concentration to the right until it concentrates entirely on the carrying infection capacity as the infection growth rate increases or the recovery rate reduces. The disease eradication and disease persistence thresholds are calculated.

We show that hyperoctahedral Whittaker functions---diagonalizing an open quantum Toda chain with one-sided boundary potentials of Morse type---satisfy a dual system of difference equations in the spectral variable. This extends a well-known bispectral duality between the nonperturbed open quantum Toda chain and a strong-coupling limit of the rational Macdonald-Ruijsenaars difference operators. It is manifest from the difference equations in question that the hyperoctahedral Whittaker function is entire as a function of the spectral variable., Comment: 20 pages, LaTeX. To appear in Int. Math. Res. Not. IMRN

Kesirli hesap teorisi, keyf mertebeden turev ve integral tanimini kapsamaktadir. Diferansiyel denklemlerin ve kesirli diferansiyel denklemlerin bazi siniflarini cozmek icin bu teori kullanilmaktadir. Bu denklemlerden birisi konfluent hipergeometrik denklemidir. Bu makalede, farkli bir cozum metodu olarak metodunun uygulanmasiyla bu denklemi cozmeyi hedeflemekteyiz.

Natural phenomena can be modelled adequately precise with differential equations. An example of such phenomena is diffusion. All differential equations with partial derivatives, of two variables and of the second order, which model the diffusion are parabolic with simple solutions. This simple solution is: erf(x), exponential functions or power series. Adding new differential terms requires the use of more complicated functions, some degenerated, modified or confluent, of real or imaginary variables. This functions are: Bessel, Kummer, Whittaker, Hermite, hyper-geometrical or Heun. The benefit of using these functions lies in greater clarity and precision in mathematical expression. Modelling methods by change-of-variable and function, adding new differential terms and transformations of special functions increase modelling possibilities of convective diffusion in different environments using differential equations. The paper presents a novel approach to diffusion modelling for dry corrosion.

In the present work, the possibility of torsional surface waves in dry sandy layer under initial stress over an inhomogeneous half-space has been carried out. The inhomogeneities in the lower semi-infinite media arise due to the polynomial variation of rigidity and density. The closed form expression for the dispersion relation of this proposed layer has been delineated by means of separation of variable method in the form of Whittaker function and its derivative retaining up to linear term. An extensive analysis has been achieved through numerical computation which explores the influence of inhomogeneity, initial stresses, sandy parameter on the torsional wave propagation have also been exhibited graphically. The obtained dispersion relation is in perfect agreement with the classical Love wave equation when derived as in particular cases. Using MATLAB software GUI has been evolved to generalize the effect of various parameters.

Nonlinear plane longitudinal elastic waves with different profiles are studied using the Murnaghan model. The novelty is that the waves are analyzed using the same approximate method and the solutions of the nonlinear wave equations are similar in form. The distortion of the initial wave profile described by cosinusoidal, Gaussian, and Whittaker functions is described theoretically and numerically. About 80 variants of initial parameters are studied numerically: three analytical representations of the initial profile, three materials (aluminum, copper, steel), three wave lengths, three initial maximum amplitudes. For each variant, four (cosine) and five (Gauss, Whittaker) two-dimensional graphs of wave shape versus traveled distance are plotted to demonstrate the distortion of the wave profile

The Present Paper devoted to investigate the effect of presence of rigid boundary on the propagation of torsional surface waves in a gravitating earth with a dry sandy medium under initial stress. The mathematical analysis of the problem has been dealt with the Whittaker function. Assuming the expansion of the Whittaker function up to linear term, it is concluded that the gravity field will always allow torsional waves to propagate in elastic and sandy medium under initial stress and rigid layer. Finally, it reveals that the sandy medium without support of a gravity field can not allow the propagation of torsional surface waves in presence of initial stress under rigid layer, where as the presence of a gravity field always supports the propagation of torsional surface waves regardless of whether the medium is elastic or dry sandy under rigid layer.

In this paper a mathematical model to study the Rayleigh wave propagation in a self-reinforced layer over an incompressible inhomogeneous elastic half-space with quadratically varying rigidity and linearly varying density has been represented. The displacement components for both the media have been deduced separately. Under suitable boundary conditions the secular equation has been established in closed form. Numerical results analyzing the equation are discussed and presented graphically. It is quite ocular that the reinforcement in the upper medium and variations in density and rigidity related to the lower semi-infinite medium has remarkable effect on the Rayleigh wave propagation. Due to the importance of self-reinforced material, the above problem has certain realistic applications in geophysics and civil engineering.

In this article, an endeavor has been made to infer the dispersion equation of the torsional surface wave in a pre-stressed an anisotropic porous medium over a dry sandy semi-limitless medium. The case insightful investigation of rigid boundary and traction free boundary surface of the upper layer has been examined in points of interest. The elastic equations of motion have been defined independently for different media utilizing Biot’s hypothesis. The dispersion equations of motion for both cases have been acquired under the suitable limit condition in the term of Whittaker function and its derivatives.

Using a recursive formula for the Mellin transform T n , a ( s ) of a spherical, principal series G L ( n , R ) Whittaker function, we develop an explicit recurrence relation for this Mellin transform. This relation, for any n ≥ 2 , expresses T n , a ( s ) in terms of a number of “shifted” transforms T n , a ( s + Σ ) , with each coordinate of Σ being a non-negative integer. We then focus on the case n = 4 . In this case, we use the relation referenced above to derive further relations, each of which involves “strictly positive shifts” in one of the coordinates of s. More specifically: each of our new relations expresses T 4 , a ( s ) in terms of T 4 , a ( s + Σ ) and T 4 , a ( s + Ω ) , where for some 1 ≤ k ≤ 3 , the kth coordinates of both Σ and Ω are strictly positive. Next, we deduce a recurrence relation for T 4 , a ( s ) involving strictly positive shifts in all three s k 's at once. (That is, the condition “for some 1 ≤ k ≤ 3 ” above becomes “for all 1 ≤ k ≤ 3 .”) These additional relations on G L ( 4 , R ) may be applied to the explicit understanding of certain poles and residues of T 4 , a ( s ) . This residue information is, as we describe below, in turn relevant to recent results concerning orthogonality of Fourier coefficients of S L ( 4 , Z ) Maass forms, and the G L ( 4 ) Kuznetsov formula.

In this paper, we study the convergence of the inverse Laplace transform for valuing American put options when the dynamics of the risky asset is governed by the constant elasticity of variance (CEV) model. The CEV model is one popular alternative of the Black-Scholes model to describe well the real financial market. We calculate various coefficients explicitly and prove that the inverse Laplace transform converges absolutely using the properties of Whittaker functions.

The present paper investigates the effect of gravity and initial stress on the propagation of torsional surface waves in heterogeneous medium. The dispersion equation has been obtained for rigid and traction free boundaries in terms of Whittaker function. The present study reveals that torsional surface waves can propagate in both the cases. In this paper, we assume the expansion of Whittaker function up to linear term. It is observed that, in presence of initial stress and gravity for both rigid and traction free boundary, the phase velocity of torsional surface waves increases with the growth of rigidity. In both the cases, it has also been noticed that as the gravity increases, the phase velocity of the torsional surface waves decreases in presence of heterogeneity and initial stress. It is concluded that in presence of traction free boundary, the phase velocity of the torsional surface waves is more than the rigid boundary.

In this sequel to our earlier works [3, 14, 15], we aim to present certain integral and series representations for special functions by using some different group theoretical methods as follows: Restrictions of the representation matrix elements to some block-diagonal matrices; Poisson transform intertwining two realizations of the SO(2, 2)-representation; Invariant properties of the bilinear integral functionals which are used to obtain the matrix elements of bases transforms operators.

We prove the irreducibility of the universal non-degenerate Whittaker modules for the affine Lie algebra sl 2 ˆ of type A 1 ( 1 ) with noncritical level. These modules can become simple Whittaker modules over sl 2 ˜ = sl 2 ˆ + C d with the same Whittaker function and central charge. We have to modulo a central character for sl 2 to obtain simple degenerate Whittaker sl 2 ˆ -modules with noncritical level. In the case of critical level the universal Whittaker module is reducible. We prove that the quotient of universal Whittaker sl 2 ˆ -module by a submodule generated by a scalar action of central elements of the vertex algebra V − 2 ( sl 2 ) is simple as sl 2 ˆ -module. We also explicitly describe the simple quotients of universal Whittaker modules at the critical level for sl 2 ˜ . Quite surprisingly, with the same Whittaker function some simple degenerate sl 2 ˜ Whittaker modules at the critical level have semisimple action of d and others have free action of d. At last, by using vertex algebraic techniques we present a Wakimoto type construction of a family of simple generalized Whittaker modules for sl 2 ˆ at the critical level. This family includes all classical Whittaker modules at critical level. We also have Wakimoto type realization for degenerate Whittaker modules for sl 2 ˆ at noncritical level.

The paper aims to study, the effect of rigid boundary on the propagation of torsional surface waves in initially stressed gravitating dry sandy half space. The mathematical analysis of the problem has been dealt with the Whittaker function. Assuming the expansion of the Whittaker function up to linear term, it is concluded that the gravity field will always allow torsional waves to propagate. The expansion of the Whittaker function up to quadratic terms shows two such wave fronts may exist in the medium. Finally, it is concluded that the sandy medium without support of a gravity field cannot allow the propagation of torsional surface waves, where as the presence of gravity field always supports the propagation of torsional surface waves regardless of whether the medium is elastic or dry sandy. Numerical results analyzing the velocity equations are discussed and presented graphically. It has been observed that density, rigidities play an important role for the propagation of torsional surface waves. It has been seen that as the non-homogeneity parameter in the layer increases, the velocity of torsional surface wave also increases. It has also been observed that an increase in compressive initial stresses decreases the velocity of torsional surface wave.

This work introduces and analyzes both the complex integration method and the modified complex integration method for two classes of highly oscillatory Bessel transforms with algebraic or algebraic-logarithmic singularities. We first transform the original integrals into the infinite integrals on [ 1 , + ∞ ) by the change of variable t = x − β , where β > 0 . With the help of the integral representation of the Bessel function, the integral representation and the asymptotic series of the Whittaker function, we then use the Cauchy’s residue theorem and convert the above infinite integrals into the infinite integrals on [ 0 , + ∞ ) . The integrands of the resulting infinite integrals are not oscillatory and decay exponentially fast. Fortunately, they can be efficiently evaluated by using the Gauss-Laguerre quadrature rule. Furthermore, we conduct their error analysis in inverse powers of the frequency ω . Finally, numerical experiments are provided to confirm our theoretical analysis. We can see from theoretical analysis and numerical experiments that the accuracy of the quadrature increases both for the case of growing the number of nodes n and fixed ω , and for the case of an increasing ω and fixed the number of nodes n . The comparison of the proposed two methods and the Filon-type method presented in Kang and Ma (2017), is briefly discussed as well.

In this paper, we study a reflected constant elasticity of variance (RCEV) process with two-sided reflecting barriers for modeling the dynamics of a foreign exchange rate in a target zone. We derive a closed form expression for the transition density by using the spectral expansion method. Monte Carlo simulation shows that our method is accurate and efficient when the results are applied to compute the expected value of the process. Finally, we illustrate that ignoring target zones in the CEV model may lead to significant computational errors.

In this paper, we construct Whittaker functions with exponential growth for the degenerate principal series of the symplectic group of genus n induced from the Siegel parabolic subgroup. This is achieved by explicitly constructing a certain Goodman–Wallach operator which yields an intertwining map from the degenerate principal series to the space of Whittaker functions, and by evaluating it on weight- l standard sections. We define a differential operator on such Whittaker functions which can be viewed as generalization of the ξ -operator on harmonic Maass forms for \SL2(\R) .

In this paper, space shift keying modulation (SSK) performance is analyzed for a $2\times 1$ multiple-input-single-output (MISO) wireless communication system with dual-hop amplify-and-forward (AF) relaying systems over mixed Rayleigh and two-wave with diffuse power (TWDP) fading channels. The cumulative distribution function (CDF) expression is derived for the end-to-end signal-to-noise ratio and it is used to evaluate the average bit error probability (ABEP) of SSK modulation in mixed fading channel. Obtained ABEP expression is in the form of Whittaker function which can be numerically evaluated using its numerical or series representation. Numerical and simulation results are presented to illustrate the impact of fading parameters on the system performance.

Purpose – The purpose of this paper is to investigate the existence of SH-waves in fiber-reinforced layer placed over a heterogeneous elastic half-space. Design/methodology/approach – The heterogeneity of the elastic half-space is caused by the exponential variations of density and rigidity. As a special case when both the layers are homogeneous, the derived equation is in agreement with the general equation of Love wave. Findings – Numerically, it is observed that the velocity of SH-waves decreases with the increase of heterogeneity and reinforced parameters. The dimensionless phase velocity of SH-waves increases with the decreases of dimensionless wave number and shown through figures. Originality/value – In this work, SH-wave in a fiber-reinforced anisotropic medium overlying a heterogeneous gravitational half-space has been investigated analytically and numerically. The dispersion equation for the propagation of SH-waves has been observed in terms of Whittaker function and its derivative of second degree order. It has been observed that on the removal of heterogeneity of half-space, and reinforced parameters of the layer, the derived dispersion equation reduces to Love wave dispersion equation thereby validates the solution of the problem. The equation of propagation of Love wave in fiber-reinforced medium over a heterogeneous half-space given by relevant authors is also reduced from the obtained dispersion relation under the considered geometry.

The present paper is to study the propagation of torsional surface wave in a non-homogeneous crustal layer over a pre-stressed dry sandy half-space. In the layer directional rigidities and density are assumed to vary quadratically with depth whereas rigidity, density and pre-stress vary linearly in the lower half-space. The dispersion equation is deducted in a closed form by means of separation of variables method in the form of Whittaker function. When the layer and the half-space both are homogeneous, the dispersion equation agrees with the equation of the classical case. The effects of non-homogeneity, pre-stress and sandy parameter on the phase velocity have also been studied graphically.

This paper introduces a new integral representation for the standard L-function of an irreducible cuspidal automorphic representation of the exceptional group G2, and also for the twist of this L-function by an arbitrary character. Because our construction unfolds to a matrix coefficient rather than a Whittaker function, it applies to non-generic representations as well as generic ones.

In the present paper, case wise studies have been made to investigate the existence of torsional surface wave in an initially stressed isotropic heterogeneous layer sandwiched between two inhomogeneous semi infinite media. Two cases have been discussed separately for wave propagation in an inhomogeneous layer between homogeneous and non-homogeneous half spaces, respectively. The dispersion relations of torsional surface wave for both the cases have been obtained in closed form. The effect of inhomogeneity parameter and initial stresses on the propagation of torsional surface wave have been discussed and presented graphically by plotting the dispersion curves.

We consider a natural basis of the Iwahori fixed vectors in the Whittaker model of an unramified principal series representation of a split semisimple p- adic group, indexed by the Weyl group. We show that the elements of this basis may be computed from one another by applying Demazure-Lusztig operators. The precise identities involve correction terms, which may be calculated by a combinatorial algorithm that is identical to the computation of the fibers of the Bott-Samelson resolution of a Schubert variety. The Demazure-Lusztig operators satisfy the braid and quadratic relations satisfied by the ordinary Hecke operators, and this leads to an action of the affine Hecke algebra on functions on the maximal torus of the L-group. This action was previously described by Lusztig using equivariant K-theory of the flag variety, leading to the proof of the Deligne-Langlands conjecture by Kazhdan and Lusztig. In the present paper, the action is applied to give a simple formula for the basis vectors of the Iwahori Whittaker functions.

We present a new framework for a broad class of affine Hecke algebra modules, and show that such modules arise in a number of settings involving representations of $p$-adic groups and $R$-matrices for quantum groups. Instances of such modules arise from (possibly non-unique) functionals on $p$-adic groups and their metaplectic covers, such as the Whittaker functionals. As a byproduct, we obtain new, algebraic proofs of a number of results concerning metaplectic Whittaker functions. These are thus expressed in terms of metaplectic versions of Demazure operators, which are built out of $R$-matrices of quantum groups depending on the cover degree and associated root system.

Let G(r) denote the metaplectic covering group of the linear algebraic group G. In this paper we study conditions on unramified representations of the group G(r) not to have a nonzero Whittaker function. We state a general Conjecture about the possible unramified characters χ such that the unramified subrepresentation of $$Ind_{{B^{\left( r \right)}}}^{{G^{\left( r \right)}}}{X^{\delta _B^{1/2}}}$$ will have no nonzero Whittaker function. We prove this Conjecture for the groups GL ( ) with r ≥ n − 1, and for the exceptional groups G 2 ( ) when r ≠ 2.

The original formulae of Kuznetsov for $SL(2,\mathbb{Z})$ allowed one to study either a spectral average via Kloosterman sums or to study an average of Kloosterman sums via a spectral interpretation. In previous papers, we have developed the spectral Kuznetsov formulae at the minimal weights for $SL(3,\mathbb{Z}))$, and in these formulae, the big-cell Kloosterman sums occur with weight functions attached to four different integral kernels, according to the choice of signs of the indices. These correspond to the $J$- and $K$-Bessel functions in the case of $GL(2)$. In this paper, we demonstrate a linear combination of the spherical and weight one $SL(3,\mathbb{Z})$ Kuznetsov formulae that isolates one particular integral kernel, which is the spherical $GL(3)$ Whittaker function. Using the known inversion formula of Wallach, we give the first arithmetic Kuznetsov formula for $SL(3,\mathbb{Z})$ and use it to study smooth averages and the Kloosterman zeta function attached to this particular choice of signs., Comment: 12 pages, retracts the meromorphic continuation of the unweighted Kloosterman zeta function

We study Whittaker functions for the principal series representations of SL(n,R)SL(n,R). We derive a system of partial differential equations characterizing our Whittaker functions. We give explicitly power series solutions at the regular singularity of the system, and integral representations of the unique moderate growth Whittaker functions.

In this paper we are concerned with numerical evaluation of Bessel transformations with the oscillators. First, we rewrite the integrals by Whittaker functions. Then, based on the asymptotic series of Whittaker functions, the integrals are transformed into the problems involving the complex exponential functions, which can be efficiently computed by using the complex integration method. At the same time, we obtain the entirely different asymptotic expansions from the references. In particular, the asymptotic expansions can be denoted by the incomplete Gamma function. The error analysis shows that the decay of the error drastically improves as the frequency grows. The effectiveness and accuracy of the methods are tested for large arguments of Bessel functions.

This paper aims to study the dispersion of torsional surface waves in a crustal layer being sandwiched between a rigid boundary plane and a sandy mantle. In the mantle, rigidity and initial stress vary linearly while density remains constant. Dispersion relation has been deduced in a closed form by means of variable separable method in the form of Whittaker function. The velocity equation for isotropic layer over a homogeneous half-space has been obtained which coincides with the standard result of Love wave under the effect of rigid boundary.