Javier Sesma

Zeros of the Hankel function of real order and of its derivative

The trajectories followed in the complex plane by all the zeros of the Hankel function and those of its derivative, when the order varies continuously along real values, are discussed.

Representation of integral dispersion relations by local forms

The representation of the usual integral dispersion relations (IDRs) of scattering theory through series of derivatives of the amplitudes is discussed, extended, simplified, and confirmed as mathematical identities. Forms of derivative dispersion relations (DDRs) valid for the whole energy interval, recently obtained and presented as double infinite series, are simplified through the use of new sum rules of the incomplete Γ functions, being reduced to single summations, where the usual convergence criteria are easily applied. For the forms of the imaginary amplitude used in phenomenology of hadronic scattering at high energies, we show that expressions for the DDR can represent, with absolute accuracy, the IDR of scattering theory, as true mathematical identities. Besides the fact that the algebraic manipulation can be easily understood, numerical examples show the accuracy of these representations up to the maximum available machine precision. As consequence of our work, it is concluded that the standard...

A new expansion of the confluent hypergeometric function in terms of modified Bessel functions

Abstract An asymptotic expansion of the confluent hypergeometric function U(a,b,x) for large positive 2a − b is given in terms of modified Bessel functions multiplied by Buchholz polynomials, a family of double polynomials in the variables b and x with rational coefficients.

Derivatives of the Pochhammer and reciprocal Pochhammer symbols and their use in epsilon-expansions of Appell and Kampé de Fériet functions

Useful expressions of the derivatives, to any order, of Pochhammer and reciprocal Pochhammer symbols with respect to their arguments are presented. They are building blocks of a procedure, recently suggested, for obtaining the ɛ-expansion of functions of the hypergeometric class related to Feynman integrals. The procedure is applied to some examples of such kind of functions taken from the literature.

A new approach to the epsilon expansion of generalized hypergeometric functions

Abstract Assuming that the parameters of a generalized hypergeometric function depend linearly on a small variable e , the successive derivatives of the function with respect to that small variable are evaluated at e = 0 to obtain the coefficients of the e -expansion of the function. The procedure, which is quite naive, benefits from simple explicit expressions of the derivatives, to any order, of the Pochhammer and reciprocal Pochhammer symbols with respect to their argument. The algorithm may be used algebraically, irrespective of the values of the parameters. It reproduces the exact results obtained by other authors in cases of especially simple parameters. Implemented numerically, the procedure improves considerably, for higher orders in e , the numerical expansions given by other methods.

Zeros of the modified Hankel function

(t) with q and x real, and x > 0. This equation occurs in certain physical problems, such as in the determination of the bound states for an inverse square potential with hard core in SchrSdinger equation. The modified Bessel function of third kind K, (z) is a regular function of z in all the z plane cut along the negative real axis. For any fixed z (z 4~ 0), K, (z) is an entire function of v. K, (z) is connected with the Hankel functions through

Two-point derivative dispersion relations

A new derivation is given for the representation, under certain conditions, of the integral dispersion relations of scattering theory through local forms. The resulting expressions have been obtained through an independent procedure to construct the real part and consist of new mathematical structures of double infinite summations of derivatives. In this new form the derivatives are calculated at the generic value of the energy E and separately at the reference point E = m that is the lower limit of the integration. This new form may be more interesting in certain circumstances and directly shows the origin of the difficulties in convergence that were present in the old truncated forms called standard-derivative dispersion relations (DDR). For all cases in which the reductions of the double to single sums were obtained in our previous work, leading to explicit demonstration of convergence, these new expressions are seen to be identical to the previous ones. We present, as a glossary, the most simplified e...

One-dimensional quantum scattering by a parabolic odd potential

Quantum scattering by a one-dimensional odd potential proportional to the square of the distance to the origin is considered. The Schrodinger equation is solved exactly and explicit algebraic expressions of the wavefunction are given. A complete discussion of the scattering function reveals the existence of Gamow (decaying) states and of resonances.

Buchholz polynomials: a family of polynomials relating solutions of confluent hypergeometric and Bessel equations

Abstract The expansion given by H. Buchholz, that allows one to express the regular confluent hypergeometric function M(a,b,z) as a series of modified Bessel functions with polynomial coefficients, is generalized to any solution of the confluent hypergeometric equation, by using a differential recurrence obeyed by the Buchholz polynomials.

An algorithm to obtain global solutions of the double confluent Heun equation

A procedure is proposed to construct solutions of the double confluent Heun equation with a determinate behaviour at the singular points. The connection factors are expressed as quotients of Wronskians of the involved solutions. Asymptotic expansions are used in the computation of those Wronskians. The feasibility of the method is shown in an example, namely, the Schrödinger equation with a quasi-exactly-solvable potential.