On applications of hypergeometric functions

An explicit numerical-analytical method is demonstrated for accurate solving the Schrödinger equation in those cases when this equation reducible to a system of n coupled ordinary differential equations with singular points. Fundamental system of solutions is constructed as algebraic combinations of power series, power functions and logarithmic function in the neighbourhood of the regular singular point and as asymptotic expansions of solutions in the neighbourhood of the irregular singular point. The method is based on the calculation of recurrent sequences of constant matrices of coefficients in power series and in inverse power series in asymptotic expansions using derived recurrent relations, that makes possible to calculate solutions at any given point using only algebraic computations and elementary functions. In turn it makes possible to solve accurately the eigenvalue problem and scattering problem and to derive analytical expressions for the wavefunctions. The method is applied to calculations of energies and wavefunctions of the discrete spectrum and wave functions of the continuous spectrum of the hydrogen-like atoms and of acceptors in semiconductors.

In this paper, a generalization of Ramanujan's cubic transformation, in the form of an inequality, is proved for zero-balanced Gaussian hypergeometric function $F(a,b;a+b;x)$, $a,b>0$.

Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions

Explicit analytical expressions to represent the radiative properties of spherical particles in the small size-parameter range were obtained. These expressions were deduced following Penndorf’s approach of expanding the Mie coefficients in power series on the size parameter. However, in opposition to Penndorf’s original work—in which some errors were found and corrected—the Mie coefficients were expanded to the eighth power of the size parameter, which results in a five-term approximation to the extinction and scattering efficiencies. Also, expressions for the evaluation of the asymmetry factor and both polarized and unpolarized phase functions were deduced and presented. The results so obtained have proved to be very accurate, even for size parameters beyond the limit of validity of the approach utilized.

The tail decay of the hitting probabilities on level zero in M/G/1-type Markov renewal processes (MRPs) is studied. The Markov renewal process is transformed into a Markov chain so that the problem of tail decay is reformulated in terms of the decay of the coefficients of a suitable power series. The latter problem is reduced to analyze the analyticity domain of the power series. Two numerical methods are introduced to compute the convergence radius of the latter power series. Some numerical experiments are presented that show the effectiveness of the proposed algorithms.

The calculation of Van der Waals forces has acquired considerable interest recently through the work of Buckingham, Knipp and others (Buckingham, 1937; Knipp, 1939). In these papers the interaction energy between two atoms is expressed as a power series in i/R, where R is the nuclear separation, and the various terms in this series are known as dipole-dipole, dipole-quadrupole, quadrupole-quadrupole, etc… interactions. In most cases only approximate values are obtainable for the coefficients in this series, though for two hydrogen atoms in their ground states, Pauling and Beach (1935) have determined the magnitudes correct to about I in 106. In this paper we discuss the simplest possible problem of this nature, i.e. the force between a bare proton and a normal unexcited hydrogen atom. We shall show that a rigorous determination of the coefficients in the power series can be made.

Our aim is to study and investigate the family of (p, q)-extended (incomplete and complete) elliptic-type integrals for which the usual properties and representations of various known results of the (classical) elliptic integrals are extended in a simple manner. This family of elliptic-type integrals involves a number of special cases and has a connection with (p, q)-extended Gauss’ hypergeometric function and (p, q)-extended Appell’s double hypergeometric function F_{1}. Turán-type inequalities including log-convexity properties are proved for these (p, q)-extended complete elliptic-type integrals. Further, we establish various Mellin transform formulas and obtain certain infinite series representations containing Laguerre polynomials. We also obtain some relationship between these (p, q)-extended elliptic-type integrals and Meijer G-function of two variables. Moreover, we obtain several connections with (p, q)-extended beta function as special values and deduce numerous differential and integral formulas. In conclusion, we introduce (p, q)-extension of the Epstein–Hubbell (E-H) elliptic-type integral.

(1958). On the Coefficients of Reciprocal Power Series. The American Mathematical Monthly: Vol. 65, No. 2, pp. 90-94.

In direct as well as diagonal reversion of a system of power series, the reversion coefficients may be expressed as polynomials in the coefficients of the original power series. These polynomials have coefficients which are natural numbers (Raney coefficients). We provide a combinatorial interpretation for Raney coefficients. Specifically, each such coefficient counts a certain collection of ordered colored trees. We also provide a simple determinantal formula for Raney coefficients which involves multinomial coefficients.

Let h* be a complex (or real) oriented cohomology theory. It has recently been observed that the of h* plays a very important role in such a theory. Above all, D. Quillen [6] showed that the formal group FU of the complex cobordism theory MU* ( ) is isomorphic to the Lazard universal formal group, and its coefficients generate the ground ring MU* (pt). Unfortunately, in the case of quaternionic oriented cohomology theory (see § 2), there exists no such formal group, since the tensor product of quaternionic line bundles does not yield a quaternionic bundle. This situation makes it difficult, for example, to produce enough generators of MSp*(pt). However there are some substitutes for the formal group. In particular, using the total Pontrjagin class of a certain quaternionic vector bundle 2C (see § 3), N. Ja. Gozman [5] defined a subring A of MU* (pt) which is contained in the image of the forgetful homomorphism (p: MSp* (pt}-*MU*(pt). In this paper, we will generalize his approach to arbitrary quaternionic oriented cohomology theory 7i*, and define a subring Ah of h* (pt) which is generated by the coefficients of certain power series. Then it will be shown that

: The usual power series representations for hypergeometric functions in two variables have a limited range of validity. In particular, they are of little use when the magnitude of one of the variables becomes very large. Using a Barnes-type integral representation and the concept of analytic continuation, the region of utility is extended to the desired domain. The poles that occur in the Barnes-type integral are assumed to be simple. Thus, explicit asymptotic expansions are obtained for each of the fourteen hypergeometric functions that belong to this class. Hypergeometric functions, Functions with a large argument, Functions of two variables, Contour integration, Power series expansions.

The Fourier Transform of the quartic Gaussian [formula omitted]: Hypergeometric functions, power series, steepest descent asymptotics and hyperasymptotics and extensions to [formula omitted

ABSTRACTAn earlier method for obtaining indefinite integrals of special function from the second-order linear equations which define them has been reformulated in terms of Riccati equations, which are nonlinear and first-order. For this application the nonlinearity is not a problem and the first-order property is a great advantage. Integrals can be derived using fragments of these Riccati equations and here only two specific fragment types are examined in detail. These fragments allow general integration formulas to be derived using quadrature. Other results will be presented separately. Results are presented here for Airy functions, Bessel functions, complete elliptic integrals, associated Legendre functions and Gauss hypergeometric functions. All results have been checked by differentiation using Mathematica.

and that the corresponding limit at x = e(O), where 0 is not congruent (mod 1) to an integral multiple of a, is zero. P. Sziisz has pointed out that for the proof of (1) and (2) Davenport's conditions can be reduced to the following: (a) limN_. EInIN ane(nt) exist for each real t, and (b) I E jnI,N a.e(nt) I <K < co uniformly in N and t. That is, the convergence of Ean and condition (6) in Davenport's paper can be replaced by (a). The purpose of this note is to generalize the above results in a couple of directions. It is to be noticed that the power series coefficients in (2), A,=g(va+y), constitute in some general sense an almost periodic sequence with the Fourier expansion

This work is motivated essentially by the fact that the applications of basic (or q-) hypergeometric functions are frequently needed in the form of summations, transformations, expansions, reductions, and integral formulas. The objective of this research paper is to study the applications of the general summation formulas contiguous to q-Kummer theorems in deriving several presumably new q-integral evaluations involving the basic (or q-) hypergeometric functions of type $$_{3} \Phi_{3}$$ and $$_{3} \Phi_{2}$$ . The results are further particularized to give specific integral evaluations in the form of summations to show the importance and utilizations of our main findings. We also derive the summation theorems for certain basic (or q-) Appell double hypergeometric functions, basic (or q-) Lauricella multiple hypergeometric functions, and certain basic (or q-) Laplace transforms by applying general summation formulas contiguous to q-Kummer theorems.