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We prove that the Weyl fractional derivative is a useful instrument to express certain properties of the zeta related functions. Specifically, we show that a known reflection property of the Hurwitz zeta function ¿(n, a) of integer first argument can be extended to the more general case of ¿(s, a), with complex s, by replacement of the ordinary derivative of integer order by Weyl fractional derivative of complex order. Besides, ¿(s, a) with (s) > 2 is essentially the Weyl (s-2)-derivative of ¿(2, a). These properties of the Hurwitz zeta function can be immediately transferred to a family of polygamma functions of complex order defined in a natural way. Finally, we discuss the generalization of a recently unveiled reflection property of the Lerch''s transcendent.

espanolSe reporta la primera cita de nidificacion conocida de pico mediano (Dendrocoptes medius, Linn. 1758) en la sierra de Elgea-Urkilla (Alava). Esta observacion tiene trascendencia para determinar mejor la distribucion, el patron de ocupacion y los factores que condicionan ambas, en el conjunto de la poblacion del Pais Vasco. euskaraOkil ertainaren (Dendrocoptes medius, Linn. 1758) lehen habiaketa ezagunaren berri eman da Elgea-Urkilla mendilerroan (Araban). Behaketa hau garrantzitsua da hobeto zehazteko EAEko biztanleriaren banaketa, okupazio eredua eta biak baldintzatzen dituzten faktoreak. EnglishIn this article we report the first known breeding record of the middle-spotted woodpecker (Dendrocoptes medius, Linn. 1758) on the Elgea-Urkilla mountain range (Alava). This observation is important as it allows us to determine the distribution and occupation patterns of this species as well as the factors that influence both of these parameters in the entire population of the Basque Country

Two decades ago, Steven Roman, Daniel E. Loeb and Gian-Carlo Rota introduced a family of harmonic numbers in their study of harmonic logarithms. We propose to refer to those numbers as {\it Roman harmonic numbers}. With the purpose of revitalizing the study of these mathematical objects, we recall here their known properties and unveil additional ones. An integral representation, several generating relations, and a collection of sum rules involving those numbers are presented. It is also shown that higher derivatives of the Pochhammer and reciprocal Pochhammer symbols are easily expressed in terms of Roman harmonic numbers., Comment: Review article

Background and Purpose— Long-term benefits of initiating stroke prophylaxis in the emergency department (ED) are unknown. We analyzed the long-term safety and benefits of ED prescription of anticoagulation in atrial fibrillation patients. Methods— Prospective, multicenter, observational cohort of consecutive atrial fibrillation patients was performed in 62 Spanish EDs. Clinical variables and thromboprophylaxis prescribed at discharge were collected at inclusion. Follow-up at 1 year post-discharge included data about thromboprophylaxis and its complications, major bleeding, and death; risk was assessed with univariate and bivariate logistic regression models. Results— We enrolled 1162 patients, 1024 (88.1%) at high risk according to CHA 2 DS 2 -VASc score. At ED discharge, 935 patients (80.5%) were receiving anticoagulant therapy, de novo in 237 patients (55.2% of 429 not previously treated). At 1 year, 48 (4.1%) patients presented major bleeding events, and 151 (12.9%) had died. Anticoagulation first prescribed in the ED was not related to major bleeding (hazard ratio, 0.976; 95% confidence interval, 0.294–3.236) and was associated with a decrease in mortality (hazard ratio, 0.398; 95% confidence interval, 0.231–0.686). Adjusting by the main clinical and sociodemographic characteristics, concomitant antiplatelet treatment, or destination (discharge or admission) did not affect the results. Conclusions— Prescription of anticoagulation in the ED does not increase bleeding risk in atrial fibrillation patients at high risk of stroke and contributes to decreased mortality.

A new representation of the Lerch's transcendent Phi(z,s,a), valid for positive integer s=n=1,2,... and for z and a belonging to certain regions of the complex plane, is presented. It allows to write an equation relating Phi(z,n,a) and Phi(1/z,n,1-a), which provides an expansion of Phi(z,n,a) as a power series of 1/z, convergent for |z|>1., 11 pages, 1 figure

The eigenstates of a real or complex cubic anharmonic oscillator are investigated using original and alternative methods. The procedure consists of determining global solutions of the Schr\"odinger equation that comply with the pertinent boundary conditions and allows us to obtain, in a very simple way, the eigenenergies and eigenfunctions of the Hamiltonian. Scattering by a real cubic potential is investigated as a specific case.

Given a second order differential equation with two singular points, namely the origin and infinity, the connection factors allow to split a power series solution into formal solutions with known asymptotic behavior. A procedure is suggested to obtain those factors, as quotients of Wronskians of the mentioned solutions, in the case of a Schrödinger equation with a polynomial potential. Application of the procedure to particular cases, whose connection factors are already known, allows us to obtain new relations for quotients and products of gamma functions.

Formal expansions, giving as particular cases semiasymptotic expansions, of the ratio of two gamma functions are obtained.

Quantum scattering by a one-dimensional odd potential proportional to the square of the distance to the origin is considered. The Schr\"odinger 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., Comment: 14 pages, 7 figures

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-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 explicit results for the DDR's in the cases of imaginary amplitudes of forms $(E/m)^\lambda[\ln (E/m)]^n$, that cover the cases of practical interest in particle physics phenomenology at high energies. We explicitly study the expressions for the cases with $\lambda$ negative odd integers, that require identification of cancelation of singularities, and provide the corresponding final results., Comment: The final publication is available at http://scitation.aip.org/content/aip/journal/jmp

Assumed that the parameters of a generalized hypergeometric function depend linearly on a small variable $\varepsilon$, the successive derivatives of the function with respect to that small variable are evaluated at $\varepsilon=0$ to obtain the coefficients of the $\varepsilon$-expansion of the function. The procedure, 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 the numerical expansions given by other methods., Comment: Some formulae added

Recently, it has been proved that a nonlinear quantum oscillator, generalization of the isotonic one, is exactly solvable for certain values of its parameters. Here we show that the Schroedinger equation for such an oscillator can be transformed into a confluent Heun equation. We give a very simple and efficient algorithm to solve it numerically, no matter what the values of the parameters are. Algebraic quasi-polynomial solutions, for particular values of the parameters, are found., 18 pages, 2 figures

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 Schroedinger equation with a quasi-exactly-solvable potential.

The z-zeros of the modified Bessel function of the third kind K_{nu}(z), also known as modified Hankel function or Macdonald function, are considered for arbitrary complex values of the order nu. Approximate expressions for the zeros, applicable in the cases of very small or very large |nu|, are given. The behaviour of the zeros for varying |nu| or arg(nu), obtained numerically, is illustrated by means of some graphics., 14 pages, 3 figures, 1 table. Added a table illustrating the numerical procedure

The representation of the usual integral dispersion relations (IDR) of scattering theory through series of derivatives of the amplitudes is discussed, extended, simplified, and confirmed as mathematical identities. Forms of derivative dispersion relations (DDR) 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 $\Gamma$ 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, 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 simplified forms sDDR, originally intended for the high energy limits, are an inconvenient and incomplete separation of terms of the full expression, leading to wrong evaluations. Since the correspondence between IDR and the DDR expansions is linear, our results have wide applicability, covering more general functions, built as combinations of well studied basic forms., Comment: 27 pages, 5 figures Few changes in text and in references To be published in Journal of Mathematical Physics

The roots of J 0 ( z ) − i J 1 ( z ) = 0 {J_0}\left ( z \right ) - i{J_1}\left ( z \right ) = 0 are saddle points of the function F 0 ( z ) ≡ z J 0 ′ ( z ) / J 0 ( z ) {F_0}\left ( z \right ) \equiv \\ z{J’_0}\left ( z \right )/{J_0}\left ( z \right ) . A very efficient algorithm allows one to obtain, by iteration, those roots to the desired accuracy.

The modulus and phase of the reduced logarithmic derivative of the Hankel function \[ z H v ( 1 ) ′ ( z ) / H v ( 1 ) ( z ) zH_v^{(1)}{}’ (z)/H_v^{(1)}(z) \] for complex variable z and real order v, are investigated. Special attention is paid to the location of saddle points and their trajectories as v varies.

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.

The modulus and phase of the reduced logarithmic derivative of the cylindrical Bessel function \[ z J ′ v ( z ) / J v ( z ) , z{J\prime _v}(z)/{J_v}(z), \] for complex variable z and real order v, are investigated. Special attention is paid to the location of saddle points and their trajectories as v varies.

To understand the origin of the difficulties in the determination of the physical wavefunc­ tion for an attractive inverse square potential, we study a model in which the singularity at the origin is substituted by a repulsive core. The structure of the spectrum of energy levels is discussed in some detail. The physical interpretation of the solutions of the Schrodinger equation for a potential of the form - (-h 2 /2m) 11/ r 2 presents difficulties, which occur for 11 larger than (l + 1/2)\ where l is the angular momentum. The difficulties are due to the fact that the condition of square integrability usually imposed on the wavefunction is not sufficient in this case to determine phase shifts or energies of bound states. For strongly attractive singular potentials, as in the case of the attractive inverse square potential, both solutions of the wave equation are square integrable, so that Von Neumann's criterium fails to determine a unique solution. A solution can be built which is finite for r~oo, but it will be finite for every value of the energy from zero to minus infinity, so that no discrete energy spectrum can be determined. The problem of the determination of the energies of the bound states for the inverse square potential has been discussed by several authors. Some of these authors postulate additional conditions on the wavefunctions representing physical states so as to obtain a discrete energy spectrum. Shortley 1 ) discussed the quantum mechanical problem as related to the classical case, and did not try to eliminate the absence of quantization of the energy levels. Landau and Lif­ shitz2) showed how the lack of quantization is related to the singularity of the potential at the origin. Recently Guggenheim, 3 ) based on simple arguments of dimensional analysis, showed how it is impossible to define uniquely energy levels for the attractive inverse square potential if no extra parameter is pro­ vided. Case, 4 ) besides assuming the square integrability of the wavefunctions, introduced the extra condition that the wavefunctions representing physical states form an orthogonal set. This assumption is sufficient to define the spacing of

These results are valid for arbitrary range and depths of the potentials here studied. In spite of the fact that for the general solution we have worked only with a particular radial dependence, for .which an explicit solution for the phase shifts can be written down, it seems plausible that the results have a more general validity. With this generalization in mind, we show that for general shapes of the radial dependence, the phase shifts in Born approximation present the momentum dependence described above. The origin of our results become transparent in this Born approximation treatment. We consider a velocity dependent potential of the form 1 )

Zeros of the Hankel function of real order follow, as the order varies continuously, trajectories across different Riemann sheets of the complex plane. A discussion of the main features of these trajectories is presented.

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 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.