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A new expression for the Hulthen off-shell Jost function in all partial waves is constructed in its maximal reduced form. As a basic requirement the on-shell solutions are first developed by following the differential equation approach to the problem together with judicious exploitation of the properties of certain special functions of mathematical physics. Utilizing the properties of the Jost function, the binding energies and phase shifts for N-N and n-d systems are computed and found excellent agreement with standard data.A new expression for the Hulthen off-shell Jost function in all partial waves is constructed in its maximal reduced form. As a basic requirement the on-shell solutions are first developed by following the differential equation approach to the problem together with judicious exploitation of the properties of certain special functions of mathematical physics. Utilizing the properties of the Jost function, the binding energies and phase shifts for N-N and n-d systems are computed and found excellent agreement with standard data.

The paper addresses the exact evaluation of the generalized Stieltjes transform Sn[f]=∫0∞f(x)(ω+x)−ndx of integral order n = 1, 2, 3, … about ω = 0 from which the asymptotic behavior of Sn[f] for small parameters ω is directly extracted. An attempt to evaluate the integral by expanding the integrand (ω + x)−n about ω = 0 and then naively integrating the resulting infinite series term by term leads to an infinite series whose terms are divergent integrals. Assigning values to the divergent integrals, say, by analytic continuation or by Hadamard’s finite part is known to reproduce only some of the correct terms of the expansion but completely misses out a group of terms. Here we evaluate explicitly the generalized Stieltjes transform by means of finite-part integration recently introduced in Galapon [Proc. R. Soc. A 473, 20160567 (2017)]. It is shown that, when f(x) does not vanish or has zero of order m at the origin such that (n − m) ≥ 1, the dominant terms of Sn[f] as ω → 0 come from contributions arising from the poles and branch points of the complex valued function f(z)(ω + z)−n. These dominant terms are precisely the terms missed out by naive term by term integration. Furthermore, it is demonstrated how finite-part integration leads to new series representations of special functions by exploiting their known Stieltjes integral representations. Finally, the application of finite part integration in obtaining asymptotic expansions of the effective diffusivity in the limit of high Peclet number, the Green-Kubo formula for the self-diffusion coefficient, and the antisymmetric part of the diffusion tensor in the weak noise limit is discussed.The paper addresses the exact evaluation of the generalized Stieltjes transform Sn[f]=∫0∞f(x)(ω+x)−ndx of integral order n = 1, 2, 3, … about ω = 0 from which the asymptotic behavior of Sn[f] for small parameters ω is directly extracted. An attempt to evaluate the integral by expanding the integrand (ω + x)−n about ω = 0 and then naively integrating the resulting infinite series term by term leads to an infinite series whose terms are divergent integrals. Assigning values to the divergent integrals, say, by analytic continuation or by Hadamard’s finite part is known to reproduce only some of the correct terms of the expansion but completely misses out a group of terms. Here we evaluate explicitly the generalized Stieltjes transform by means of finite-part integration recently introduced in Galapon [Proc. R. Soc. A 473, 20160567 (2017)]. It is shown that, when f(x) does not vanish or has zero of order m at the origin such that (n − m) ≥ 1, the dominant terms of Sn[f] as ω → 0 come from contributions arisin...

We obtain a new family of coherent state representations of SU(n+1), in which the coherent states are Wigner functions over a subgroup of SU(n+1). For representations of SU(n+1) of the type (λ, 0, 0,…), the basis functions are simple products of n exponential. The corresponding coherent state representations of the algebra su(n+1) are also obtained, and provide a polar decomposition of su(n+1) for any n+1. The su(n+1) modules thus obtained are useful in understanding contractions of su(n+1) and su(n+1)-phase states of quantum optics.

Symmetric informationally complete positive operator-valued measures (SIC-POVMs) have many applications in quantum information. However, it is not easy to construct SIC-POVMs and there are only a few known classes of them, and we do not even know whether there exists an infinite class of them, thus constructing approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs) has its own meaning. In this paper, we use character sums over finite fields to present two constructions of ASIC-POVMs. We show that there are some classes of infinite families of ASIC-POVMs by using some special functions over finite fields.

As a basis for electronic structure calculations, Gaussians are inconvenient because they show unsuitable behavior at larger distances, while Hankel functions are singular at the origin. This paper discusses a new set of special functions which combine many of the advantageous features of both families. At large distances from the origin, these “smoothed Hankel functions” resemble the standard Hankel functions and therefore show behavior similar to that of an electronic wave function. Near the origin, the functions are smooth and analytical. Analytical expressions are derived for two-center integrals for the overlap, the kinetic energy, and the electrostatic energy between two such functions. We also show how to expand such a function around some point in space and discuss how to evaluate the potential matrix elements efficiently by numerical integration. This supplies the elements needed for a practical application in an electronic structure calculation.

Canonical quantization of the isomonodromy solutions of equations integrable via the Inverse Scattering Transform leads to generalized Knizhnik–Zamolodchikov equations. One can solve these equations by the off-shell Bethe ansatz method provided the Knizhnik–Zamolodchikov equations are related with the highest weight representations of the corresponding Lie algebras: These solutions can be written in terms of multi-variable generalizations of special functions of the hypergeometric type. In this work, we consider a realization of the above scheme for the Maxwell–Bloch system with pumping: quantum states for this system are found in terms of the multi-variable confluent hypergeometric function.

Quadratic form approach allows for new results in the analysis of a class of integral-difference operators in finite domains: non-negativity, spectral estimations, a new property of Legendre polynomials, and establishing links with weighted mean-square deviation functionals and with infinite Jacobi matrices with not-bounded coefficients. Generalisation of integral-difference operators to higher dimensions is provided and application to matter relaxation in a field is considered. A new class of special functions naturally appears.

A general method is presented to determine the symmetry operators of linear q‐difference equations. It is applied to q‐difference analogs of the Helmoltz, heat, and wave equations in diverse dimensions. The symmetries are found to generate q‐deformations of classical Lie algebras. The method of separation of variables is used to obtain solutions which are seen to involve many basic special functions. This allows the derivation of various identities and formulas for these q‐functions.

Quantum canonical transformations are used to derive the integral representations and Kummer solutions of the confluent hypergeometric and hypergeometric equations. Integral representations of the solutions of the non-periodic three body Toda equation are also found. The derivation of these representations motivate the form of a two-dimensional generalized hypergeometric equation which contains the non-periodic Toda equation as a special case and whose solutions may be obtained by quantum canonical transformation., Comment: LaTeX, 24 pp., Imperial-TP-93-94-5 (revision: two sections added on the three-body Toda problem and a two-dimensional generalization of the hypergeometric equation)

Caustics arising in the asymptotic description of rapidly decaying solutions of the equations integrable via the inverse scattering transform are defined. Different asymptotic approaches to the description are considered. The appearance of members (Ρn2) of the hierarchy of the second Painleve equation as special functions of wave catastrophies is discussed. The ‘‘adjoining’’ problem Ρn2 → Ρk2(n≳k) for the simplest example Ρ22 → Ρ2 is considered in detail.

The localization behavior of one‐dimensional quantum systems for ℏ→0 is investigated by semiclassical methods. In particular the localization of the quantum probability around turning points of arbitrary even order associated to classical hyperbolic orbits is considered and a relation of the localization speed in ℏ with the classical motion is established. Our analysis is based on local norm comparisons of solutions to Schrodinger type equations; it relies mainly on a combination of scaling and asymptotic arguments and thus evades the use of special functions. Applications of the results to separable multidimensional Schrodinger equations are indicated by a brief discussion of the one‐electron diatomic molecular ion.

The on‐shell tree amplitudes of 2‐D P‐adic strings are constructed. Along with the recently obtained on‐shell tree amplitudes of the ordinary (archimedean) 2‐D string, they lead to adelic product formulas for all N‐point amplitudes. This is in marked difference from 26‐D critical strings, where such adelic formulas only obtain for four‐point amplitudes. Certain much‐discussed operator redefinitions are considered at the P‐adic level where they change radically, the P‐adic gamma function having only one rather than infinitely many real poles. Also considered are q‐strings in 2‐D.

A quantum three‐body system with zero angular momentum is considered. The plane wave F related to this problem is studied. It is proved to be a function of two variables that have a meaning of the eikonals on the internal space. A number of explicit formulas for F and its asymptotics are derived. New addition theorems for the scalar hyperspherical harmonics as well as for some special functions are obtained.

A given unitary representation of the group SL(2,R), belonging to the discrete series, is shown to involve necessarily some special functions (in particular, Laguerre and Hardy–Pollaczek polynomials). Various realizations of this representation are investigated, including the coherent states one. More generally, it is shown that the representations of the discrete series of the universal covering of SL(2,R) involves generalized Laguerre and Pollaczek polynomials. The Riemann zeta function is shown to be concerned with these representations.

An exactly solvable problem for the finite‐difference Schrodinger equation in the relativistic configurational space is considered. The appropriate finite‐difference generalization of the factorization method is developed. The theory of new special functions ‘‘the relativistic Hermite polynomials,’’ in which the solutions are expressed, is constructed.

Bounded and π‐periodic eigenfunctions of the operator d2/dv2+[2(μ−ν +(μ+ν+1)cos 2v)/sin 2v] (d/dv)+8ξ cos 2v+4γ2 sin2 2v are studied. These solutions are new special functions, particular cases of which will be the spheroidal, the Coulomb spheroidal, and the hyperspheroidal harmonics. The algebraic technique due to the quantum inverse scattering method is applied to obtain the three‐term recursion relations for the coefficients of eigenfunctions expansion at the Jacobi polynomials series and hence the suitable equation (the continued fraction is equal to zero) for eigenvalues. The special functions considered generalize the Truskova polyspheroidal periodic functions (γ=0).

The conditions for superintegrable systems in two-dimensional Euclidean space admitting separation of variables in an orthogonal coordinate system and a functionally independent third-order integral are studied. It is shown that only systems that separate in subgroup type coordinates, Cartesian or polar, admit potentials that can be described in terms of nonlinear special functions. Systems separating in parabolic or elliptic coordinates are shown to have potentials with only non-movable singularities.

The Schwinger-Keldysh nonequilibrium theory allows the description of various transport phenomena involving bosons (fermions) embedded in bosonic (fermionic) environments. The retarded Green’s function obeys the Dyson equation and determines via its non-vanishing asymptotic behavior the dissipationless open dynamics. The appearance of this regime is conditioned by the existence of the solution of a general class of transcendental equations in complex domain that we study. Particular cases consist in transcendental equations containing exponential, hyperbolic, power law, logarithmic, and special functions. The present analysis provides an analytical description of the thermal and temporal correlation function of two general observables of a quantum system in terms of the corresponding spectral function. Special integral properties of the spectral function guarantee non-vanishing asymptotic behavior of the correlation function.

Nonlinear conservation laws driven by Levy processes have solutions which, in the case of supercritical nonlinearities, have an asymptotic behavior dictated by the solutions of the linearized equations. Thus, the explicit representation of the latter is of interest in the nonlinear theory. In this paper, we concentrate on the case where the driving Levy process is a multiscale stable (anomalous) diffusion, which corresponds to the case of multifractal conservation laws considered in Biler et al. [J. Differ. Equations 148, 9 (1998); Stud. Math. 135, 231 (1999); Ann. Inst. Henri Poincare, Anal. Nonlineaire 18, 613 (2001); and Stud. Math. 148, 171 (2001)]. The explicit representations, building on the previous work on single-scale problems (see, e.g., Gorska and Penson [Phys. Rev. E 83, 061125 (2011)]), are developed in terms of the special functions (such as Meijer G functions) and are amenable to direct numerical evaluations of relevant probabilities.

The 2D space-fractional Schrodinger equation in the time-independent and time-dependent cases for the scattering problems in the fractional quantum mechanics is studied. We define the Green's functions for the two cases and give the mathematical expression of them in infinite series form and in terms of some special functions. The asymptotic formulas of the Green's functions are also given, and applied to get the approximate wave functions for the fractional quantum scattering problems. These results contain those in the standard (integer) quantum mechanics as special cases, and can be applied to study the complex quantum systems.

The use of the umbral formalism allows a significant simplification of the derivation of sum rules involving products of special functions and polynomials. We rederive in this way known sum rules and addition theorems for Bessel functions. Furthermore, we obtain a set of new closed form sum rules involving various special polynomials and Bessel functions. The examples we consider are relevant for applications ranging from plasma physics to quantum optics., Comment: J. Math. Phys. vol. 54, 073501 (2013), 6 pages

In a recent study of representing Dirac's delta distribution using q-exponentials, Jauregui and Tsallis experimentally discovered formulae for π as hypergeometric series as well as certain integrals. Herein, we offer rigorous proofs of these identities using various methods and our primary intent is to lay down an illustration of the many technical underpinnings of such evaluations. This includes an explicit discussion of creative telescoping and Carlson's Theorem. We also generalize the Jauregui–Tsallis identities to integrals involving Chebyshev polynomials. In our pursuit, we provide an interesting tour through various topics from classical analysis to the theory of special functions.

We provide with a generalization of well known (p,q)-deformed Heisenberg algebras, called (R,p,q)-deformed quantum algebras, and study the corresponding (R,p,q)-series. A general formulation of the binomial theorem is given. Special functions are obtained as limit cases. This work well prolongs a previous work by Odzijewicz [Commun. Math. Phys. 192, 183 (1998)]. Known results in the literature are recovered.

Properties of the four families of recently introduced special functions of two real variables, denoted here by $E^\pm$, and $\cos^\pm$, are studied. The superscripts $^+$ and $^-$ refer to the symmetric and antisymmetric functions respectively. The functions are considered in all details required for their exploitation in Fourier expansions of digital data, sampled on square grids of any density and for general position of the grid in the real plane relative to the lattice defined by the underlying group theory. Quality of continuous interpolation, resulting from the discrete expansions, is studied, exemplified and compared for some model functions., Comment: 22 pages, 10 figures

We obtain several new closed-form expressions for the evaluation of a family of infinite-domain integrals of the Whittaker functions Wκ,μ(x) and Mκ,μ(x) and the modified Bessel functions Iμ(x) and Kμ(x) with respect to the index μ. The new family of definite integrals is useful in a variety of contexts in mathematical physics. In particular, the integral involving Kμ(x) represents a new example of the Kontorovich–Lebedev transform. We discuss the relationship between the results derived here and the previously known integrals of Whittaker and Bessel functions. In some cases, we obtain entirely new expressions, and in other cases, we generalize previously known results. An application to time-dependent radiation transport theory is also discussed.

We investigate the Hamiltonian as well as Lax formulations of the polytropic gas dynamics which can be characterized by the polytropic exponent γ=1+h. We show that when h=1∕m with ∣m∣∊N⩾2, the resonant phenomena occur in bi-Hamiltonian formulation and logarithmic-type conserved charges emerge naturally. We provide a logarithmic Lax representation for conserved charges and the whole hierarchy flows, which coincides with the bi-Hamiltonian formulation constructed from a two-dimensional Frobenius manifold associated with the polytropic gas dynamics.

We find sets of solutions to the generalized spheroidal wave equation (GSWE) or, equivalently, to the confluent Heun equation. Each set is constituted by three solutions, one given by a series of ascending powers of the independent variable, and the others by series of regular and irregular confluent hypergeometric functions. For a fixed set, the solutions converge over different regions of the complex plane but present series coefficients proportional to each other. These solutions for the GSWE afford solutions to a double-confluent Heun equation by a taking-limit process due to Leaver. [E. W. Leaver, J. Math. Phys. 27, 1238 (1986)]. Another procedure, called Whittaker-Ince limit [B. D. Figueiredo, J. Math. Phys. 46, 113503 (2005)], provides solutions in series of powers and Bessel functions for two other equations with a different type of singularity at infinity. In addition, new solutions are obtained for the Whittaker-Hill and Mathieu equations [F. M. Arscott, Proc. R. Soc. Edinburg A67, 265 (1967)] by considering these as special cases of both the confluent and double-confluent Heun equations. In particular, we find that each of the Lindemann-Stieltjes solutions for the Mathieu equation [E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press (1945)] is associated with two expansions in series of Bessel functions. We also discuss a set of solutions in series of hypergeometric and confluent hypergeometric functions for the GSWE and use their Leaver limits to obtain infinite-series solutions for the Schrodinger equation with an asymmetric double-Morse potential. Finally, the possibility of extending the solutions of the GSWE to the general Heun equation is briefly discussed.

We consider sums over the square lattice which depend only on radial distance, and provide formulas which enable sums of functions with Neumann series to be reexpressed as combinations of hypergeometric series. We illustrate the procedure using trigonometric sums previously studied by Borwein and Borwein, sums combining logarithms, Bessel functions Jλ, and powers of distance, and sums of Neumann functions. We also exhibit sums which may be evaluated analytically and recurrence formulas linking sums.

A comprehensive study of the symmetric Levy stable probability density function is presented. This is performed for orders both less than 2, and greater than 2. The latter class of functions are traditionally neglected because of a failure to satisfy non-negativity. The complete asymptotic expansions of the symmetric Levy stable densities of order greater than 2 are constructed, and shown to exhibit intricate series of transcendentally small terms—asymptotics beyond all orders. It is demonstrated that the symmetric Levy stable densities of any arbitrary rational order can be written in terms of generalized hypergeometric functions, and a number of new special cases are given representations in terms of special functions. A link is shown between the symmetric Levy stable density of order 4, and Pearcey’s integral, which is used widely in problems of optical diffraction and wave propagation. This suggests the existence of applications for the symmetric Levy stable densities of order greater than 2, despite their failure to define a probability density function.

A class of spherical functions is studied which can be viewed as the matrix generalization of Bessel functions. We derive a recursive structure for these functions. We show that they are only special cases of more general radial functions which also have a properly generalized, recursive structure. Some explicit results are worked out. For the first time, we identify a subclass of such radial functions which consist of a finite number of terms only.

The analytic and asymptotic properties of the spherically symmetric d-dimensional Levy stable probability density function, pαd(r), are discussed in detail. These isotropic stable probability density functions (pdfs) are analogous to the one-dimensional symmetric Levy stable pdfs previously studied by the present authors [J. Math. Phys. 43, 2670 (2002)]. We construct a hypergeometric representation of pαd(r) when α is rational, and find a number of new representations of pαd(r) in terms of special functions for various values of d and α. A recursion relation is found between pαd(r) and pαd+2(r), which, in particular, implies there exists a simple map between pα1(r) and pα3(r). As in our previous paper, we discuss the properties of pαd(r) for both the cases α⩽2 and α>2. We demonstrate the existence of intricate exponentially small series in the large r asymptotics of pαd(r) when α is an integer, which are dominant when α is even. We explicitly construct this beyond all orders expansion of pαd(r) for arbitr...

Asymptotic limits are given for the SU(2) Wigner Dmnj functions as j→∞ for three domains of m and n. Similar asymptotic limits are given for the SU(3) Wigner functions of an irrep with highest weight (λ,0) as λ→∞. The results are shown to be relevant to the analysis of experiments with quantum interferometers.

P functions and confluent P (CP) functions are classified into two and five groups respectively according to the types of the step operators (SO’s) intrinsic to the respective classes of functions. The correspondence between the types of the SO’s and the realizations of the Lie algebras G (a,b) and T6 is established as follows. The modified SO’s acting on P functions (SOP’s) belong to either of the type A and E realizations of G (1,0) and T6 respectively. The modified SOC’s, namely the SO’s acting on CP functions, belong to one of the type B, C′, C″, F, D′ realizations of G (1,0), G (0,1), G (0,0), or T6.

The mathematical formalism recently developed for solving multiterm linear recursion relations is used to obtain a solution of the linear potential problem, for any value of the orbital quantum number l, in terms of combinatorics functions. The wavefunctions are given as power series expansions and the energy eigenvalues as the roots of an infinite order polynomial.