Kurt Siegfried Kolbig

The polygamma function p ( k ) ( x ) for x= 1 4 and x= 3 4

Expressions for the polygamma function @j^(^k^)(x) for the arguments x=14 and x=14 are given in terms of Bernoulli numbers, Euler numbers, the Riemann zeta function for odd integer arguments, and the related series of reciprocal powers of integers @b(m).

The complete Bell polynomials for certain arguments in terms of Stirling numbers of the first kind

Abstract The value of the (exponential) complete Bell polynomials for certain arguments, given essentially by finite sums of reciprocal powers with a real parameter, is expressed in terms of coefficients generated by the power series expansion of a Pochhammer symbol. This result generalizes an earlier result given by Comtet, which relates these polynomials for certain arguments to Stirling numbers of the first kind.

Remarks on the computation of Coulomb wavefunctions.

Abstract Recent work on the computation of Coulomb wavefunctions is reviewed and references are given.

The polygamma function Ψ (k) (x) for x=¼ and x=¾

Expressions for the polygamma function @j^(^k^)(x) for the arguments x=14 and x=14 are given in terms of Bernoulli numbers, Euler numbers, the Riemann zeta function for odd integer arguments, and the related series of reciprocal powers of integers @b(m).

Some special cases of the generalized hypergeometric function q+1 F q

Abstract Some special cases of the generalized hypergeometric function q +1 F q with rational numbers as parameters are given in tabular form. These results complement existing tables. Some analytical aspects are discussed, and a derivation is given for those cases which correct existing table entries or replace numerical values by analytic expressions.

On the Zeros of the Incomplete Gamma Function

Some asymptotic formulae given elsewhere for the zeros of the incomplete gam- ma function -y(a, x) are corrected. A plot of a few of the zero trajectories of the function y(xw, x) is given, where x is a real parameter. Based on theoretical work by Mahler, it is seen that the zero trajectories of y(xw, x) lie in a finite region of the complex w-plane. 1. Introduction. Let a = a + 43 be a complex variable. The incomplete gamma function Py(a, x) can then be defined by (1) n=O n! a+n The zeros of this function have been treated theoretically by several authors, for example, by Franklin (2), Walther (3), Rasch (4), Hille and Rasch (5), Mahler (6), and Tricomi (7), (8), (9); some of the papers seem to be little known. Numerical values for complex zeros have been given by Franklin (2) for -y(a, 1), and more recently by Kolbig (10), (11), who calculated the first eight zero trajectories of y(a, x) in the a-plane as functions of x > 0. It is the aim of this note to recall some of the earlier results (occasionally cor- recting them), and to present a plot containing a few of the zero trajectories of the function -y(xw, x) in the complex w-plane (w = u + iv), as functions of the real parameter x > 0. It will be seen that these trajectories all lie in a finite region of the w-plane, and that they cluster towards a limiting curve as shown by Mahler (6), who investigated the function y(xw, x) in detail.

On a Hankel transform integral containing an exponential function and two Laguerre polynomials

A Hankel transform integral of a product of a power, an exponential function and two Laguerre polynomials is discussed. Formulae for this integral are given in several tables, but most of them contain the same error, i.e. a pair of subscripts has been interchanged. We derive here the correct expression.

Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function

Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function are calculated as functions of the upper limit X of the definition integrals. It becomes apparent that not all, but only some, of the zero trajectories of the incomplete Riemann zeta function join a zero of the Riemann zeta function 1(s) for X -+ C. The remaining trajectories at least in the region considered, approach the zero trajectories of the incomplete gamma function.

Some infinite integrals with powers of logarithms and the complete Bell polynomials

Modern computing tools, such as Computer Algebra, often allow a straightforward evaluation of mathematical expressions, for example by using recurrence relations. However, results so obtained may hide structures, which in some cases are not immediately recognized. This is discussed for a definite integral that is related to the higher derivatives of the gamma function. Two other, similar integrals are also considered.

On the integral ∫ ∞ 0 X v-1 (1+βX) -λ ln m X dX

A recurrence relation is given for the integral in the title. Formulae which allows easy evaluation by formula manipulation on a computer are given for integer and half-integer values of v and λ. A comparison with formulae given in integral tables is made. Using results obtained, four infinite series are expressed in closed form, generalizing well-known results for infinite series of reciprocal powers. Tables of explicit expressions for the above integral for small values of m, with v = n, v = n + 12, λ = l, λ = l + 12, and certain values of β, are also presented.