Confluent Hypergeometric Series Research Articles

Generalized Dirichlet Distribution Based on Confluent Hypergeometric Series

Dirichlet distribution is a kind of high-dimensional continuous probability distribution, which has important applications in the fields of statistics, machine learning and bioinformatics. In this paper, based on gamma distribution we study two two-dimensional random variables. Then we derive the properties of these two two-dimensional random variables by using the properties of non-central gamma distribution and confluent hypergeometric series. From these properties, we find the two random variables follow generalized Dirichlet distributions. Applying hypergeometric series to Dirichlet distribution broadens the research of Dirichlet distribution.

Branched continued fraction representations of ratios of Horn's confluent function $\mathrm{H}_6$

In this paper, we derive some branched continued fraction representations for the ratios of the Horn's confluent function $\mathrm{H}_6.$ The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. We establish the estimates of the rate of convergence for the branched continued fraction expansions in some region $\Omega$ (here, region is a domain (open connected set) together with all, part or none of its boundary). It is also proved that the corresponding branched continued fractions uniformly converge to holomorphic functions on every compact subset of some domain $\Theta,$ and that these functions are analytic continuations of the ratios of double confluent hypergeometric series in $\Theta.$ At the end, several numerical experiments are represented to indicate the power and efficiency of branched continued fractions as an approximation tool compared to double confluent hypergeometric series.

The Quantum Mechanics Canonically Associated to Free Probability I: Free Momentum and Associated Kinetic Energy

After a short review of the quantum mechanics canonically associated with a classical real valued random variable with all moments, we begin to study the quantum mechanics canonically associated to the standard semi-circle random variable [Formula: see text], characterized by the fact that its probability distribution is the semi-circle law [Formula: see text] on [Formula: see text]. We prove that, in the identification of [Formula: see text] with the [Formula: see text]-mode interacting Fock space [Formula: see text], defined by the orthogonal polynomial gradation of [Formula: see text], [Formula: see text] is mapped into position operator and its canonically associated momentum operator [Formula: see text] into [Formula: see text] times the [Formula: see text]-Hilbert transform [Formula: see text] on [Formula: see text]. In the first part of the present paper, after briefly describing the simpler case of the [Formula: see text]-harmonic oscillator, we find an explicit expression for the action, on the [Formula: see text]-orthogonal polynomials, of the semi-circle analogue of the translation group [Formula: see text] and of the semi-circle analogue of the free evolution [Formula: see text], respectively, in terms of Bessel functions of the first kind and of confluent hyper-geometric series. These results require the solution of the inverse normal order problem on the quantum algebra canonically associated to the classical semi-circle random variable and are derived in the second part of the present paper. Since the problem to determine, with purely analytic techniques, the explicit form of the action of [Formula: see text] and [Formula: see text] on the [Formula: see text]-orthogonal polynomials is difficult, the above mentioned results show the power of the combination of these techniques with those developed within the algebraic approach to the theory of orthogonal polynomials.

Turań-type inequalities for two hypergeometric supertrigonometric functions associated with Kummer confluent hypergeometric series of first type

In the paper, on account of the Kummer confluent hypergeometric series of first type, we suggest the Turań-type inequalities for the hypergeometric supersine and supercosine functions by using the Cauchy–Bunyakovsky–Schwarz inequality.

An extension of Pochhammer’s symbol and its application to hypergeometric functions, II

Recently we have introduced a productive form of gamma and beta functions and applied them for generalized hypergeometric series [Filomat, 31 (2017), 207-215]. In this paper, we define an additive form of gamma and beta functions and study some of their general properties in order to obtain a new extension of the Pochhammer symbol. We then apply the new symbol for introducing two different types of generalized hypergeometric functions. In other words, based on the defined additive beta function, we first introduce an extension of Gauss and confluent hypergeometric series and then, based on two additive types of the Pochhammer symbol, we introduce two extensions of generalized hypergeometric functions of any arbitrary order. The convergence of each series is studied separately and some illustrative examples are given in the sequel.

An extension of Pochhammer’s symbol and its application to hypergeometric functions

By using a special property of the gamma function, we first define a productive form of gamma and beta functions and study some of their general properties in order to define a new extension of the Pochhammer symbol. We then apply this extended symbol for generalized hypergeometric series and study the convergence problem with some illustrative examples in this sense. Finally, we introduce two new extensions of Gauss and confluent hypergeometric series and obtain some of their general properties.

Recursion formulas for the Srivastava–Daoust and related multivariable hypergeometric functions

This paper concludes the study of recursion formulas of multivariable hypergeometric functions. Earlier in [V. Sahai and A. Verma, Recursion formulas for multivariable hypergeometric functions, Asian–Eur. J. Math. 8 (2015) 50, 1550082], the authors have given the recursion formulas for three variable Lauricella functions, Srivastava’s triple hypergeometric functions and [Formula: see text]-variable Lauricella functions. Further, in [V. Sahai and A. Verma, Recursion formulas for Recursion formulas for Srivastava’s general triple hypergeometric functions, Asian–Eur. J. Math. 9 (2016) 17, 1650063], we have obtained recursion formulas for Srivastava general triple hypergeometric function [Formula: see text]. We present here the recursion formulas for generalized Kampé de Fériet series and Srivastava and Daoust multivariable hypergeometric function. Certain particular cases leading to recursion formulas of certain generalized hypergeometric function of one variable, certain Horn series, Humbert’s confluent hypergeometric series and some confluent forms of Lauricella series in [Formula: see text]-variables are also presented.

Orthogonal polynomials on the unit circle satisfying a second-order differential equation with varying polynomial coefficients

ABSTRACTConsider the linear second-order differential equation(1.1) where with or , are polynomials with complex coefficients and . Under some assumptions over a certain class of lowering and raising operators, we show that for a sequence of polynomials orthogonal on the unit circle to satisfy the differential equation (1.1), the polynomial must be of a specific form involving and extension of the Gauss and confluent hypergeometric series.

Elementary functions in thermodynamic Bethe ansatz**Based on talks given at ‘Infinite Analysis 2014’ (Tokyo, 2014) and at ‘Integrable lattice models and quantum field theories’ (Bad Honnef, 2014).

Some years ago, Fendley found an explicit solution to the thermodynamic Bethe ansatz (TBA) equation for an supersymmetric theory in 2D with a specific F-term. Motivated by this, we seek explicit solutions for other super-potential cases utilizing the idea from the ODE/IM correspondence. We find that the TBA equations, corresponding to a wider class of super-potentials, admit solutions in terms of elementary functions such as modified Bessel functions and confluent hyper-geometric series.

A new algorithm for internal heat generation in nanofluid flow due to a stretching sheet in a porous medium

Purpose – The purpose of this paper is to study heat and mass transfer in copper-water and silver-water nanofluid flow over stretching sheet placed in saturated porous medium with internal heat generation or absorption. The authors further introduce a new algorithm for solving heat transfer problems in fluid mechanics. The model used for the nanofluid incorporates the nanoparticle volume fraction parameter and a consideration of the chemical reaction effects among other features. Design/methodology/approach – The partial differential equations for heat and mass transfer in copper-water and silver-water nanofluid flow over stretching sheet were transformed into a system of nonlinear ordinary differential equations. Exact solutions for the boundary layer equations were obtained in terms of a confluent hypergeometric series. A novel spectral relaxation method (SRM) is used to obtain numerical approximations of the governing differential equations. The exact solutions are used to test the convergence and accuracy of the SRM. Findings – Results were obtained for the fluid properties as well as the skin friction, and the heat and mass transfer rates. The results are compared with limiting cases from previous studies and they show that the proposed technique is an efficient numerical algorithm with assured convergence that serves as an alternative to numerical methods for solving nonlinear boundary value problems. Originality/value – A new algorithm is used for the first time in this paper. In addition, new exact solutions for the energy and mass transport equations have been obtained in terms of a confluent hypergeometric series.

CONFLUENT GAMMA DENSITY IN MODELLING TSUNAMI INTEREVENT TIMES.

We consider here a probability model which associates the usual gamma form with a confluent hypergeometric series. The probability model is termed as confluent gamma density. Some distributional properties of this model are derived. A graphical representation of the model for the varying values of parameter is illustrated. The model thrusts into an interesting application which relates it with tsunami event modelling. A study on tsunami occurrence is essential since it happens frequently at different parts of the world. Here we give an overall view about tsunamis and analyze tsunami interevent times.

The Bivariate Confluent Hypergeometric Series Distribution and Some of Its Properties

In this paper we develop a bivariate version of the confluent hypergeometric series distribution through its probability generating function and study some of its properties by deriving its probability mass function, factorial moments, probability generating functions of its marginal and conditional distributions and recursion formulae for probabilities, raw moments and factorial moments. Further certain mixtures and limiting cases of this distribution are also obtained.

Revised Line Profile Function for Hydrogenic Species

Abstract Analytical series expansions for the hydrogenic spectral line profile functions are derived starting from the three single expressions, obtained by integrating twice the convolution of the Holtsmark, Lorentz and Doppler line profile functions. We get well converging series expansions for the line wings and centers by reducing the number of arguments in the profile function by one, introducing the module of the Holtsmark and Lorentz profiles as a new argument. In the intermediate part of the line, the parabolic cylinder functions expressed via the confluent hypergeometric series, are used. The multi-component Stark splitting of the hydrogenic spectral lines and the modeled stochastic electron transitions in the electric field of the adjacent ions generate wide Doppler plateaux at the line centers, with the characteristic widths estimated from the fit to the characteristic width of the Holtsmark profile. This additional Doppler broadening of the line profile function removes the central dip typical to the Holtsmark profile.

HYPERGEOMETRIC BERNOULLI POLYNOMIALS AND APPELL SEQUENCES

There are two analytic approaches to Bernoulli polynomials Bn(x): either by way of the generating function zexz/(ez - 1) = ∑ Bn(x)zn/n! or as an Appell sequence with zero mean. In this article, we discuss a generalization of Bernoulli polynomials defined by the generating function zN exz/(ez - TN-1(z)), where TN(z) denotes the Nth Maclaurin polynomial of ez, and establish an equivalent definition in terms of Appell sequences with zero moments in complete analogy to their classical counterpart. The zero-moment condition is further shown to generalize to Bernoulli polynomials generated by the confluent hypergeometric series.

Exact evaluation of maximal-ratio and equal-gain diversity receivers for M-ary QAM on Nakagami fading channels

Exact integral expressions are derived for calculating the symbol-error rate (SER) of multilevel quadrature amplitude modulation (MQAM) in conjunction with L-fold antenna diversity on arbitrary Nakagami fading channel. Both maximal-ratio combining (MRC) (in independent and correlated fading) and equal-gain combining (EGC) predetection (in independent fading) diversity techniques have been considered. Exact closed-form SER expressions for two restricted Nakagami fading cases (MRC reception) are also derived. An exact analysis of EGC for MQAM has not been reported previously, despite its practical interest. Remarkably, the exact SER integrals can also be replaced by a finite-series approximation formula. A useful procedure for computing the confluent hypergeometric series is also presented.

Functional equation underlying the quantum theory of fields

In the case of equality of vector and axial-vector couplings, useful to investigate the problem of mass generation for spinor fields by spontaneous symmetry breaking, all one-loop two-point Green's functions appear finite. The Mellin-transformed Schwinger-Dyson integral equations reduce to a finite-difference equation solved by a confluent hypergeometric function of the reduced kinematic variable. In this paper, the limiting process from the massive spinor to the massless case is treated in detail, all subsidiary terms being bounded and expressible in terms of Appell series and generalized confluent hypergeometric series.

Partial Recovery of Loss of Array Gain due to Quantization in Digital Beamformers by Post Filtering

By assuming a sinusoidal wave embedded in Gaussian noise as array input signal, the output autocorrelation function of a digital beamformer can be expressed as the sum of infinite series Rsxs, Rnxn, and Rsxn, involving signal and its harmonics, ascending powers of noise cross-correlation functions, and “quantizer functions” of various orders [H. S. C. Wang, J. Acoust. Soc. Am. 53, 929 (1973)]. It is demonstrated that the effect of bandpass filtering of post-beamforming signal can be rigorously analyzed, without transforming to the frequency domain, by applying certain factors attributed first to Davenport [W. B. Davenport, Jr., J. Appl. Phys. 24, 702 (1953)] to the successive terms of the series Rnxk and Rsxn-In the special case of DIMUS processor, the series of quantizer functions reduce exactly to Davenport's confluent hyper-geometric series for his “bandpass rooter” with n → ∞. For a 12-channel beamformer, the array gain improvement or partial recovering of quantizing (clipping) loss by post filtering in correlated and uncorrelated noise fields at different values of input SNR is numerically evaluated and displayed.

An Analysis of the Properties of the Exact Finite Sample Distribution of a Nonconsistent GCL Structural Variance Estimator

An asymptotic expansion of a confluent hypergeometric series is used to approximate the exact finite sample distribution function of a nonconsistent GCL structural variance estimator. A theoretical result is used to motivate the specification of a simple algorithm under which we may accept or reject the use of the asymptotic distribution function of the GCL estimator to approximate the exact distribution function.

An Asymptotic Expansion for the Distribution of the Eigenvalues of a 3 by 3 Wishart Matrix

A parametrization of the rotation group $O^+(p)$ of $p$ by $p$ orthogonal matrices with determinant $+1$ in terms of their skew symmetric parts is used to derive, for $p = 3$, an explicit expansion for $_0F_0^{(p)}(Z, \Omega)$, a hypergeometric function of two matrix arguments appearing in the distribution of the eigenvalues of a $p$ by $p$ Wishart matrix. On the basis of a numerically derived simplification of the low order terms of this series, an asymptotic expansion of $_0F_0^{(3)}$ in terms of products of ordinary confluent hypergeometric series is conjectured. Limited numerical exploration indicates the new series to be several orders of magnitude more accurate than the series from which it was derived.

Transformed hypergeometric transmission lines

A transformation procedure is described for determining a set of nonautonomic distributions which are complementary to previously described general classes of lines. A table of these distributions is presented, together with the conditions necessary for their evaluation. In particular, for the Gaussian distributed RC line, which is inherent, for example, in some diffused-semiconductor realisations, a solution is obtained in terms of a convergent confluent hypergeometric series which is equivalent to an earlier solution in terms of Hermite polynomials. Finally, its performance is evaluated and experimentally compared with a well known distribution.