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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.

An algorithm is proposed for obtaining global solutions of the biconfluent Heun equation, which appears when dealing with a variety of physical problems. The procedure, which provides algebraic expressions of the solutions in the form of convergent series or asymptotic expansions, lies on the determination of the connection factors relating the solutions about the regular singular point at the origin and the irregular one at infinity. The algorithm is illustrated by examples.

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.

The Laplace transform representation of the incomplete beta function Bx (a,b) obtain a very simple asymptotic expansion, in ascending powers of 1/a, whose truncation error can be easily estimated.

The zeros in the complex z plane of the Whittaker function W c/z,μ (z), closely related to spherical waves in the quantum-mechanical Coulomb problem, are investigated for varying real values of the parameters c and μ.

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.

(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

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.